Ali, Jake, Jessica, Kayla Walk the Path and Find Destination. Measure displacement. Graphical Results Percent Error for Graphical Results Analytical Results Percent Error for Analytical Results
Shoot the Grade 10/26
Ali, Jessica, Kayla, Jake Purpose Our objective is to launch a ball from the launcher at an angle of 10 degrees and speed (6.92 m/s) so that the ball passes consecutively through five rings and lands in a cup on the floor. We are trying to see if our calculations will correctly place the hoops in the right place so that the ball will effectively project through. We have to factor in the pull of gravity (when concerning the projectile) at a force of -9.8 m/s/s and air resistance. Prediction/ Hypothesis The ball will launch successfully through the hoops based on the heights that we had calculated. The rings will begin by increasing in height and then eventuate into a shorter height, replicating a parabolic trajectory. Materials launcher, ball, carbon paper, masking tape, measuring tape, right angle clamps, plumb bomb, string Procedure By using our typical projectile equations, we were able to calculate where the hoops should exactly be placed. After calculating, we left the notebook and went to the real world scenario: setting up tape hoops via ceiling tiles and string We then measured the heights which were calculated, for both the y and x axises, and placed each hoop in their designated location. In order to ensure that our markings were correct, we launched the ball each time we set up a hoop. After each trial, we would maneuver the hoop slightly to make sure the ball would go through. Eventually, with a little manipulation, we got through setting up all hoops so that the ball would successfully project through all, technically landing in the cup.
BELOW is the video of the ball shooting thru four hoops
Observations and Data Finding initial velocity was necessary when determining where to place the hoops. Below are our calculations.
finding average distance
finding Vit
finding time
initial velocity
Chart for finding distance and time to place hoops, relative to our initial height of 1.202.
Analysis below is a chart showing percent error, with a sample underneath. Conclusion We thought that the ball would be able to go through all of the hoops, because we felt that our calculations were very accurate; however, it ended up that our calculations were slightly off. From our calculations, we got that the initial velocity would be about 6.92 m/s. Based on this calculated value, we found two distance components of where to place the hoops: vertical and horizontal. As shown in the data above, our values were the following- hoop 1: 1.2742 m vertically and .7185 meters horizontally, hoop 2: 1.2374 m vertically and 1.437 m horizontally, hoop 3: 1.0924 m vertically and 2.156 m horizontally, hoop4: .83350 m vertically and 2.874 m horizontally, and lastly hoop 5: .47440 m vertically and 3.593 m horizontally. It turned out that our horizontal values did not have to be altered at all, but the vertical heights differed. We were able to manipulate the hoops slightly so that the ball would go thru the hoops, with the heights recorded above. If we had time we most likely could have got the last hoop to the right height so that the ball would launch through it. As you can see above, our percent error in each case is very small, suggesting that our original calculations were somewhat accurate. Sources to account for our error were the following: Since there were many groups setting up the hoops, often, the strings were moved, and in some cases they fell, forcing our group to start setting it up again. Another source of error was the angle of the launcher. Often, after several trials, the angle would slightly increase from 10 degrees to around 15 degrees. Despite constant checking for this error, it kept occurring. We couldn't be as accurate as possible. One way to change the lab to address the first source of error is to make sure that everyone sets up their launcher in separate locations. Therefore, there will not be any issue with a group altering another's set up. Another way to change the lab to address the second source of error would be to have an item that can hold the launcher in place (regarding its angle). It seemed that the screws on the side that were designed to hold the launcher in place were not 100% effective. Thus, finding another object to keep it in place would be the best solution. This can be applied to many real life situations, ranging from some careers to simple observations. It is good to know the logistics behind a projectile and to understand how it functions because you never know when you may need the information; it may be something as simple as calculating how to score a field goal or how to hit the tennis ball into the right spot on the court. It's also beneficial that physics can help in these situations, all with the result of a logical solution. This lab showed us the importance of trial and error and the importance of consistency.
Class Notes
Intro to Vectors
Magnitude: the value without the sign
scalar is a quantitative value
ie-speed, time, mass, distance
Vector: quantity with both magnitude and direction
components are the perpendicular pieces added to make the vector
ie- velocity, acceleration, displacement, force
Resultant: single vector that replaces the components collinear: same line noncollinear: different line
biggest possible magnitude is adding, and the smallest is subtracting. these are our max and min.
Vector Addition
Graphical Methods The Analytical Method
Horizontally Launched Projectiles
We use the same equations we used with 1D kinematics
a= Vf- Vi/ Δt
Vf= Vi +aΔt (no displacement)
Δd= 1/2( Vi + Vf)t
Δd= Vit + 1/2at(squared)
Vf(squared)=Vi(squared)+2aΔd
Sample:
Ground to Ground
the Y displacement ALWAYS equals ZERO Systems Problem
Off the Cliff
Activities
Launching Ball Activity (10/21)
Ali, Jake, Kayla, Jessica How fast does the launcher shoot the ball at medium range? Change the initial height, calculate where to place the cup on the floor so that the ball lands inside of it 3 times in a row. Below is the procedure we used to get the ball in the cup. As you can see, the ball entered the cup, but due to a lack of strength of the tape, it fell over. However, it did enter the cup, but rolled out.
Percent Error
Gourdarama Project
Ali and Nicole
our project, at 1.3 kg
Conclusions: Our velocity was 5.263 m/s and our acceleration was -2.77 m/s/s. Our gourd traveled 5 meters and weighed about 1.3 kg (a very light weight, not to mention!!) If Nicole and I were to change anything, we would have probably made sure that the axiles were lined up straighter so that the mobile would not crash into the wall. By doing this, the rate of "acceleration" would have decreased because it would have covered a longer distance. As you can tell, the high rate of acceleration was our biggest problem. By cutting this rate down, our cart would have been more successful. We could have also used three wheels instead of four to cut down on the weight a little bit more!
Homework
Vectors Lesson 1 a +b (10/12)
Vectors can only be described by magnitude and direction. An arrow diagram is used to display vector form, and can even be used as a quantitative value, where you can multiply, divide, subtract, and add vectors. Vectors and Direction A vector quantity is a quantity that is fully described by both magnitude and direction. While a scalar quantity is a quantity that is dfully described by its magnitude. Examples of vector quantities displacement, velocity, acceleration, and force. Vector quantities are not fully described unless both magnitude and direction are listed. Vector quantities are often represented by scaled vector diagrams.They depict a vector by use of an arrow drawn to scale in a specific direction. They are sometimes called free-body diagrams. Below are some characteristics:
a scale is clearly listed
a vector arrow is drawn in a specified direction. The vector arrow has a head and a tail.
the magnitude and direction of the vector is clearly labeled
Below are some examples of what this will look like:
Vector Addition Two vectors can be added together to determine the result. The net force experienced by an object is determined by computing the vector sum of all the individual forces acting upon that object. Example:
The Pythagorean Theorem The Pythagorean theorem determines the result of adding ONLY two vectors that make a RIGHT angle to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
To see how the method works, consider the following problem: Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.
The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km.
Using Trigonometry to Determine a Vector's Direction The direction of a resultant vector can often be determined by use of trigonometric functions. Sine, cosine, and tangent relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. Below are the equations.
These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement.
Thus, trig functions can give us degree, but not always direction. We sometimes need to figure out direction ourselves.
Use of Scaled Vector Diagrams to Determine a Resultant The head-to-tail method is used to determine the vector sum. An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors: 20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m
The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram. SCALE: 1 cm = 5 m
FUN FACT: The order in which three vectors are added has no affect upon either the magnitude or the direction of the resultant. The result will still have the same magnitude and direction.
Vectors Lesson 1 c +d (10/13)
Vectors have resultants and components. Resultants are the *result* of the addition of two or more vectors. Components are the 2-dimensial breakdown of making up a vector.
Resultants
The resultant is the vector sum of two or more vectors. If displacement vectors A, B, and C are added together, the result will be vector R.
Displacement vector R gives the same result as displacement vectors A + B + C. That is why it can be said that
A + B + C = R
The above discussion pertains to the result of adding displacement vectors. When displacement vectors are added, the result is a resultant displacement.
Vector Components
A vector is a quantity that has both magnitude and direction. When there was a free-bodydiagram depicting the forces acting upon an object, each individual force was directed in one dimension - either up or down or left or right. When an object had an acceleration and we described its 1-D direction. Now, we will have two dimensions - upward and rightward, northward and westward, eastward and southward, etc
In situations in which vectors are directed at angles to the customary coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes. For example, a vector that is directed northwest can be thought of as having two parts - a northward part and a westward part.
Each part of a two-dimensional vector is known as a component.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference.
Any vector directed in two dimensions can be thought of as having two different components. The component of a single vector describes the influence of that vector in a given direction.
Vectors Lesson 1 e (10/17)
The resultants of vectors can be measured using two methods: the parallelogram method and the trigonomic method. Both require drawing diagrams and come out with similar results.
Vector Resolution
A vector directed in two dimensions can be thought of as having two components.
The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution that we will examine are
the parallelogram method
the trigonometric method
Parallelogram Method of Vector Resolution
The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. If one desires to determine the components as directed along the traditional x- and y-coordinate axes, then the parallelogram is a rectangle with sides that stretch vertically and horizontally.
Step by step breakdown:
Select a scale and accurately draw the vector to scale in the indicated direction.
Sketch a parallelogram around the vector: beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a rectangle (a special case of a parallelogram).
Draw the components of the vector. The components are the sides of the parallelogram. The tail of the components start at the tail of the vector and stretches along the axes to the nearest corner of the parallelogram. Be sure to place arrowheads on these components to indicate their direction
Meaningfully label the components of the vectors with symbols to indicate which component represents which side
Measure the length of the sides of the parallelogram and use the scale to determine the magnitude of the components in real units. Label the magnitude on the diagram.
EXAMPLE:
Trigonometric Method of Vector Resolution The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. Trigonometric functions relate the ratio of the lengths of the sides of a right triangle to the measure of an acute angle within the right triangle. They can be used to determine the length of the sides of a right triangle if an angle measure and the length of one side are known.
Step by step breakdown:
Construct a rough sketch (no scale needed) of the vector in the indicated direction. Label its magnitude and the angle that it makes with the horizontal.
Draw a rectangle about the vector such that the vector is the diagonal of the rectangle. Beginning at the tail of the vector, sketch vertical and horizontal lines. Then sketch horizontal and vertical lines at the head of the vector. The sketched lines will meet to form a rectangle.
Draw the components of the vector. The components are the sides of the rectangle. The tail of each component begins at the tail of the vector and stretches along the axes to the nearest corner of the rectangle. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right)
Meaningfully label the components of the vectors with symbols to indicate which component represents which side
To determine the length of the side opposite the indicated angle, use the sine function. Substitute the magnitude of the vector for the length of the hypotenuse. Use some algebra to solve the equation for the length of the side opposite the indicated angle
Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.
EXAMPLE:
Vectors Lesson 1f (10/18)
Component Method of Vector Addition When the two vectors are added head-to-tail as shown below, the resultant is the hypotenuse of a right triangle. A right triangle has two sides plus a hypotenuse; so the Pythagorean theorem is perfect for adding two right angle vectors.
Addition of Three or More Right Angle Vectors Example 1: A student drives his car 6.0 km, North before making a right hand turn and driving 6.0 km to the East. Finally, the student makes a left hand turn and travels another 2.0 km to the north. What is the magnitude of the overall displacement of the student?
The head-to-tail vector addition diagram is shown below.
The order in which the three vectors are added must be changed in order to make a right triangle. If the three vectors are added in the order 6.0 km, N + 2.0 km, N + 6.0 km, E, then the diagram will look like this:
After rearranging the order in which the three vectors are added, the resultant vector is now the hypotenuse of a right triangle. The lengths of the perpendicular sides of the right triangle are 8.0 m, North (6.0 km + 2.0 km) and 6.0 km, East. The magnitude of the resultant vector (R) can be determined using the Pythagorean theorem. R2 (8.0 km)2 + (6.0 km)2 R2 64.0 km2+ 36.0 km2 R2 100.0 km2 R SQRT (100.0 km2) R = 10.0 km Adding vectors A + B + C gives the same resultant as adding vectors B + A + C or even C + B + A. As long as all three vectors are included with their specified magnitude and direction, the resultant will be the same.
The resultant of the addition of three or more right angle vectors can be easily determined using the Pythagorean theorem. Doing so involves the adding of the vectors in a different order.
SOH CAH TOA and the Direction of Vectors The convention is known as the counter-clockwise from east convention, often abbreviated as the CCW convention. Using this convention, the direction of a vector is often expressed as a counter-clockwise angle of rotation of the vector about its tail from due East.
Example: 6.0 km, N + 6.0 km, E + 2.0 km, N. In this problem, we know the length of the side opposite theta (Θ) - 6.0 km - and the length of the side adjacent the angle theta (Θ) - 8.0 km. The tangent function will be used to calculate the angle measure of theta (Θ). Tangent(Θ) = Opposite/Adjacent Tangent(Θ) = 6.0/8.0 Tangent(Θ) = 0.75 Θ = tan-1 (0.75) Θ = 36.869 …° Θ =37°
The result is 37° east of north. Since the angle that the resultant makes with east is the complement of the angle that it makes with north, we could express the direction as 53° CCW.
In summary, the direction of a vector can be determined by finding the angle of rotation counter-clockwise from due east.
Addition of Non-Perpendicular Vectors It is possible to force two (or more) non-perpendicular vectors to be transformed into other vectors that do form a right triangle. The trick involves the concept of a vector component and the process of vector resolution. Example: Max plays middle linebacker for South's football team. During one play in last Friday night's game against New Greer Academy, he made the following movements after the ball was snapped on third down. First, he back-pedaled in the southern direction for 2.6 meters. He then shuffled to his left (west) for a distance of 2.2 meters. Finally, he made a half-turn and ran downfield a distance of 4.8 meters in a direction of 240° counter-clockwise from east (30° W of S) before finally knocking the wind out of New Greer's wide receiver. Determine the magnitude and direction of Max's overall displacement.
As is the usual case, the solution begins with a diagram of the vectors being added.
The resultant is the vector sum of these three vectors; a head-to-tail vector addition diagram reveals that the resultant is directed southwest. Of the three vectors being added, vector C is clearly the nasty vector. Its direction is neither due south nor due west. The solution involves resolving this vector into its components.
Vector C makes a 30° angle with the southern direction. By sketching a right triangle with horizontal and vertical legs and C as the hypotenuse, it becomes possible to determine the components of vector C. This is shown in the diagram below. The side adjacent this 30° angle in the triangle is the vertical side; the vertical side represents the vertical (southward) component of C - Cy. So to determine Cy, the cosine function is used. The side opposite the 30° angle is the horizontal side; the horizontal side represents the horizontal (westward) component of C – C. The cosine function is used to determine the southward component since the southward component is adjacent to the 30° angle. The sine function is used to determine the westward component since the westward component is the side opposite to the 30° angle.
Now our vector addition problem has been transformed from the addition of two nice vectors and one nasty vector into the addition of four nice vectors.
With all vectors oriented along are customary north-south and east-west axes, they can be added head-to-tail in any order to produce a right triangle whose the hypotenuse is the resultant.
The triangle's perpendicular sides have lengths of 4.6 meters and 6.756 meters. The length of the horizontal side (4.6 m) was determined by adding the values of B (2.2 m) and Cx (2.4 m). The length of the vertical side (6.756… m) was determined by adding the values of A (2.6 m) and Cy (4.156… m). The resultant's magnitude (R) can now be determined using the Pythagorean theorem. R2 (6.756… m)2 + (4.6 m)2 R2 45.655… m2 + 21.16 m2 R2 66.815… m2 R SQRT(66.815… m2 ) R = 8.174 … m R = ~8.2 m
tangent(Θ) = (6.756… m)/(4.6 m) = 1.46889… so… Θ = tan-1 (1.46889…) = 55.7536… ° Θ = ~56° This 56° SW angle is the angle between the resultant vector . The (CCW) can be determined by adding 180° to the 56°. So the CCW direction is 236°.
Example: Cameron Per (his friends call him Cam) and Baxter Nature are on a hike. Starting from home base, they make the following movements. A: 2.65 km, 140° CCW B: 4.77 km, 252° CCW C: 3.18 km, 332° CCW
This is the angle measure diagrams.
A table to organize values.
Vector
East-West Component
North-South Component
A 2.65 km 140° CCW
(2.65 km)•cos(40°) = 2.030… km, West
(2.65 km)•sin(40°) = 1.703… km, North
B 4.77 km 252° CCW
(4.77 km)•sin(18°) = 1.474… km, West
(4.77 km)•cos(18°) = 4.536… km, South
C 3.18 km 332° CCW
(3.18 km)•cos(28°) = 2.808… km, East
(3.18 km)•sin(28°) = 1.493… km, South
Sum of A + B + C
0.696 km, West
4.326 km, South
R2 (0.696 km)2 + (4.326 km)2 R2 0.484 km2 + 18.714 km2 R2 19.199 km2 R SQRT(19.199 km2) R = ~4.38 km
Tangent(Θ) = opposite/adjacent Tangent(Θ) = (4.326 km)/(0.696 km) Tangent(Θ) = 6.216 Θ = tan-1(6.216) Θ = 80.9° It would be worded as 80.9° SW. Since west is 180° counterclockwise from east, the direction could also be expressed in the counterclockwise (CCW) from east convention as 260.9°.
FINAL RESULTANT: 4.38 km with a direction of 260.9° (CCW).
Vectors Lesson 1 g+h (10/18)
On occasion objects move within a medium that is moving with respect to an observer. In such instances, the magnitude of the velocity of the moving object (whether it be a plane or a motorboat) with respect to the observer on land will not be the same as the speedometer reading of the vehicle. Motion is relative to the observer. The observer on land, often named (or misnamed) the "stationary observer" would measure the speed to be different than that of the person on the boat. The observed speed of the boat must always be described relative to who the observer is. To illustrate this principle, consider a plane flying amidst a tailwind. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity. The resultant velocity of the plane (that is, the result of the wind velocity contributing to the velocity due to the plane's motor) is the vector sum of the velocity of the plane and the velocity of the wind. This resultant velocity is quite easily determined if the wind approaches the plane directly from behind. If the plane encounters a headwind, the resulting velocity will be less than 100 km/hr. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a side wind of 25 km/hr, West. The resulting velocity of the plane is the vector sum of the two individual velocities. To determine the resultant velocity, the plane velocity (relative to the air) must be added to the wind velocity. This is the same procedure that was used above for the headwind and the tailwind situations; only now, the resultant is not as easily computed. Since the two vectors to be added - the southward plane velocity and the westward wind velocity - are at right angles to each other, the Pythagorean can be used. In this situation of a side wind, the southward vector can be added to the westward vector using the usual methods of vector addition. The magnitude of the resultant velocity is determined using Pythagorean theorem. The direction of the resulting velocity can be determined using a trigonometric function. Since the plane velocity and the wind velocity form a right triangle when added together in head-to-tail fashion, the angle between the resultant vector and the southward vector can be determined using the sine, cosine, or tangent functions. The tangent function can be used. Like any vector, the resultant's direction is measured as a counterclockwise angle of rotation from due East. The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat. If a motorboat were to head straight across a river, it would not reach the shore directly across from its starting point. The river current influences the motion of the boat and carries it downstream. Motorboat problems such as these are typically accompanied by three separate questions:
What is the resultant velocity (both magnitude and direction) of the boat?
If the width of the river is X meters wide, then how much time does it take the boat to travel shore to shore?
What distance downstream does the boat reach the opposite shore?
The first of these three questions: the resultant velocity of the boat can be determined using the Pythagorean theorem (magnitude) and a trigonometric function (direction). The second and third of these questions can be answered using the average speed equation.
ave. speed = distance/time. A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. That is to say, if you pull upon an object in an upward and rightward direction, then you are exerting an influence upon the object in two separate directions - an upward direction and a rightward direction. These two parts of the two-dimensional vector are referred to as components. A component describes the affect of a single vector in a given direction. Any force vector that is exerted at an angle to the horizontal can be considered as having two parts or components. The vector sum of these two components is always equal to the force at the given angle. Any vector - whether it is a force vector, displacement vector, velocity vector, etc. - directed at an angle can be thought of as being composed of two perpendicular components. These two components can be represented as legs of a right triangle formed by projecting the vector onto the x- and y-axis.The two perpendicular parts or components of a vector are independent of each other. All vectors can be thought of as having perpendicular components. In fact, any motion that is at an angle to the horizontal or the vertical can be thought of as having two perpendicular motions occurring simultaneously. These perpendicular components of motion occur independently of each other. Any component of motion occurring strictly in the horizontal direction will have no affect upon the motion in the vertical direction. Any alteration in one set of these components will have no affect on the other set.
Vectors Lesson 2 a+b (10/19)
Lesson A Central idea: A projectile is an object upon which the only force is gravity. Gravity causes a vertical motion, which causes a vertical acceleration.
What is a projectile?
It is an object dropped from rest or thrown vertically upwards at an angle to the horizontal
It is any object projected or dropped that continues in motion by its own inertia
What forces affect upon a projectile?
The only force is of downward gravity
What is inertia?
The resistance an object has to a change in a state of motion
How do we represent projectiles?
A free-body diagram is a good representation of projectiles.
What type of motion will we see from this diagram?
We will see a Parabolic Trajectory, similar to a diagram in free fall.
Lesson B Central idea: A projectile has two components: a vertical motion and horizontal motion. Vertical motion is effected by the downward force of gravity, and therefore is always changing. However, horizontal motion is unaffected by gravity, and therefore stays at a constant rate.
Can projectiles move in a direction other than vertical motion?
Yes, projectiles can move in a horizontal motion as well. Horizontal and vertical are both COMPONENTS of the projectile.
Does gravity affect the horizontal motion of a projectile?
No, it does not. This is because gravity acts as a downward force, therefore unable to affect the horizontal motion.
How does a projectile travel?
With constant horizontal velocity and a downward vertical acceleration.
Is the velocity of downward vertical motion changing or constant?
It is ALWAYS changing, due to the account of gravity's acceleration of -9.8 m/s/s.
Is acceleration present in horizontal motion?
No. It is always moving at a constant rate because it is not affected by gravity.
Vectors Lesson 2 c (10/20)
Central Idea: Depending on the projectile, it can either be moving against time solely, or considering the property of displacement.Both of these properties can be addressed numerically.
What is the velocity of horizontal trajectory? vertical?
Constant and changing, respectively.
What is the difference between free fall and a projectile? What are the equations for projectiles?
Free fall is an example of projectiles. However, in many cases, a projectile will have an initial velocity (as opposed to zero velocity) due to a launch or upward motion.
What are the equations for certain projectiles?
vertical displacement for a 1D object
y= 1/2gt2
vertical displacement for a horizontally launched projectile
X= Vixt
vertical displacement for an angled-launched projectile
viyt+ 1/2gt2
Can vertical motion increase? decrease?
Yes, the diagram should most of the time look like a parabola. Therefore, the velocities will increase and decrease.
What is the difference between a launched projectile over time versus one dealing with displacement?
If displacement is a factor, the numbers will be increasing or decreasing (uniformed rate) for horizontal components.
Table of Contents
Labs
Cafeteria Vector Lab 10/19
Ali, Jake, Jessica, KaylaWalk the Path and Find Destination. Measure displacement.
Graphical Results
Percent Error for Graphical Results
Analytical Results
Percent Error for Analytical Results
Shoot the Grade 10/26
Ali, Jessica, Kayla, JakePurpose
Our objective is to launch a ball from the launcher at an angle of 10 degrees and speed (6.92 m/s) so that the ball passes consecutively through five rings and lands in a cup on the floor. We are trying to see if our calculations will correctly place the hoops in the right place so that the ball will effectively project through. We have to factor in the pull of gravity (when concerning the projectile) at a force of -9.8 m/s/s and air resistance.
Prediction/ Hypothesis
The ball will launch successfully through the hoops based on the heights that we had calculated. The rings will begin by increasing in height and then eventuate into a shorter height, replicating a parabolic trajectory.
Materials
launcher, ball, carbon paper, masking tape, measuring tape, right angle clamps, plumb bomb, string
Procedure
By using our typical projectile equations, we were able to calculate where the hoops should exactly be placed. After calculating, we left the notebook and went to the real world scenario: setting up tape hoops via ceiling tiles and string We then measured the heights which were calculated, for both the y and x axises, and placed each hoop in their designated location. In order to ensure that our markings were correct, we launched the ball each time we set up a hoop. After each trial, we would maneuver the hoop slightly to make sure the ball would go through. Eventually, with a little manipulation, we got through setting up all hoops so that the ball would successfully project through all, technically landing in the cup.
BELOW is the video of the ball shooting thru four hoops
Observations and Data
Finding initial velocity was necessary when determining where to place the hoops. Below are our calculations.
Chart for finding distance and time to place hoops, relative to our initial height of 1.202.
Analysis
below is a chart showing percent error, with a sample underneath.
Conclusion
We thought that the ball would be able to go through all of the hoops, because we felt that our calculations were very accurate; however, it ended up that our calculations were slightly off. From our calculations, we got that the initial velocity would be about 6.92 m/s. Based on this calculated value, we found two distance components of where to place the hoops: vertical and horizontal. As shown in the data above, our values were the following- hoop 1: 1.2742 m vertically and .7185 meters horizontally, hoop 2: 1.2374 m vertically and 1.437 m horizontally, hoop 3: 1.0924 m vertically and 2.156 m horizontally, hoop4: .83350 m vertically and 2.874 m horizontally, and lastly hoop 5: .47440 m vertically and 3.593 m horizontally. It turned out that our horizontal values did not have to be altered at all, but the vertical heights differed. We were able to manipulate the hoops slightly so that the ball would go thru the hoops, with the heights recorded above. If we had time we most likely could have got the last hoop to the right height so that the ball would launch through it. As you can see above, our percent error in each case is very small, suggesting that our original calculations were somewhat accurate. Sources to account for our error were the following: Since there were many groups setting up the hoops, often, the strings were moved, and in some cases they fell, forcing our group to start setting it up again. Another source of error was the angle of the launcher. Often, after several trials, the angle would slightly increase from 10 degrees to around 15 degrees. Despite constant checking for this error, it kept occurring. We couldn't be as accurate as possible. One way to change the lab to address the first source of error is to make sure that everyone sets up their launcher in separate locations. Therefore, there will not be any issue with a group altering another's set up. Another way to change the lab to address the second source of error would be to have an item that can hold the launcher in place (regarding its angle). It seemed that the screws on the side that were designed to hold the launcher in place were not 100% effective. Thus, finding another object to keep it in place would be the best solution. This can be applied to many real life situations, ranging from some careers to simple observations. It is good to know the logistics behind a projectile and to understand how it functions because you never know when you may need the information; it may be something as simple as calculating how to score a field goal or how to hit the tennis ball into the right spot on the court. It's also beneficial that physics can help in these situations, all with the result of a logical solution. This lab showed us the importance of trial and error and the importance of consistency.
Class Notes
Intro to Vectors
Magnitude: the value without the sign
- scalar is a quantitative value
- ie-speed, time, mass, distance
Vector: quantity with both magnitude and direction- components are the perpendicular pieces added to make the vector
- ie- velocity, acceleration, displacement, force
Resultant: single vector that replaces the componentscollinear: same line
noncollinear: different line
biggest possible magnitude is adding, and the smallest is subtracting. these are our max and min.
Vector Addition
Graphical MethodsThe Analytical Method
Horizontally Launched Projectiles
We use the same equations we used with 1D kinematics- a= Vf- Vi/ Δt
- Vf= Vi +aΔt (no displacement)
- Δd= 1/2( Vi + Vf)t
- Δd= Vit + 1/2at(squared)
- Vf(squared)=Vi(squared)+2aΔd
Sample:Ground to Ground
the Y displacement ALWAYS equals ZEROSystems Problem
Off the Cliff
Activities
Launching Ball Activity (10/21)
Ali, Jake, Kayla, JessicaHow fast does the launcher shoot the ball at medium range?
Change the initial height, calculate where to place the cup on the floor so that the ball lands inside of it 3 times in a row.
Below is the procedure we used to get the ball in the cup. As you can see, the ball entered the cup, but due to a lack of strength of the tape, it fell over. However, it did enter the cup, but rolled out.
Percent Error
Gourdarama Project
Ali and NicoleConclusions: Our velocity was 5.263 m/s and our acceleration was -2.77 m/s/s. Our gourd traveled 5 meters and weighed about 1.3 kg (a very light weight, not to mention!!)
If Nicole and I were to change anything, we would have probably made sure that the axiles were lined up straighter so that the mobile would not crash into the wall. By doing this, the rate of "acceleration" would have decreased because it would have covered a longer distance. As you can tell, the high rate of acceleration was our biggest problem. By cutting this rate down, our cart would have been more successful. We could have also used three wheels instead of four to cut down on the weight a little bit more!
Homework
Vectors Lesson 1 a +b (10/12)
Vectors can only be described by magnitude and direction. An arrow diagram is used to display vector form, and can even be used as a quantitative value, where you can multiply, divide, subtract, and add vectors.Vectors and Direction
A vector quantity is a quantity that is fully described by both magnitude and direction. While a scalar quantity is a quantity that is dfully described by its magnitude.
Examples of vector quantities displacement, velocity, acceleration, and force. Vector quantities are not fully described unless both magnitude and direction are listed.
Vector quantities are often represented by scaled vector diagrams.They depict a vector by use of an arrow drawn to scale in a specific direction. They are sometimes called free-body diagrams.
Below are some characteristics:
- a scale is clearly listed
- a vector arrow is drawn in a specified direction. The vector arrow has a head and a tail.
- the magnitude and direction of the vector is clearly labeled
Below are some examples of what this will look like:Vector Addition
Two vectors can be added together to determine the result. The net force experienced by an object is determined by computing the vector sum of all the individual forces acting upon that object.
Example:
The Pythagorean Theorem
The Pythagorean theorem determines the result of adding ONLY two vectors that make a RIGHT angle to each other. The Pythagorean theorem is a mathematical equation that relates the length of the sides of a right triangle to the length of the hypotenuse of a right triangle.
To see how the method works, consider the following problem:
Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.
The result of adding 11 km, north plus 11 km, east is a vector with a magnitude of 15.6 km.
Using Trigonometry to Determine a Vector's Direction
The direction of a resultant vector can often be determined by use of trigonometric functions. Sine, cosine, and tangent relate an acute angle in a right triangle to the ratio of the lengths of two of the sides of the right triangle. Below are the equations.
These three trigonometric functions can be applied to the hiker problem in order to determine the direction of the hiker's overall displacement.
Thus, trig functions can give us degree, but not always direction. We sometimes need to figure out direction ourselves.
Use of Scaled Vector Diagrams to Determine a Resultant
The head-to-tail method is used to determine the vector sum.
An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors:
20 m, 45 deg. + 25 m, 300 deg. + 15 m, 210 deg.SCALE: 1 cm = 5 m
The head-to-tail method is employed as described above and the resultant is determined (drawn in red). Its magnitude and direction is labeled on the diagram.
SCALE: 1 cm = 5 m
FUN FACT: The order in which three vectors are added has no affect upon either the magnitude or the direction of the resultant. The result will still have the same magnitude and direction.
Vectors Lesson 1 c +d (10/13)
Vectors have resultants and components. Resultants are the *result* of the addition of two or more vectors. Components are the 2-dimensial breakdown of making up a vector.Resultants
The resultant is the vector sum of two or more vectors. If displacement vectors A, B, and C are added together, the result will be vector R.
Displacement vector R gives the same result as displacement vectors A + B + C. That is why it can be said that
A + B + C = R
The above discussion pertains to the result of adding displacement vectors. When displacement vectors are added, the result is a resultant displacement.
Vector Components
A vector is a quantity that has both magnitude and direction. When there was a free-bodydiagram depicting the forces acting upon an object, each individual force was directed in one dimension - either up or down or left or right. When an object had an acceleration and we described its 1-D direction. Now, we will have two dimensions - upward and rightward, northward and westward, eastward and southward, etc
In situations in which vectors are directed at angles to the customary coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes. For example, a vector that is directed northwest can be thought of as having two parts - a northward part and a westward part.
Each part of a two-dimensional vector is known as a component.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components - an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body - one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference.
Any vector directed in two dimensions can be thought of as having two different components. The component of a single vector describes the influence of that vector in a given direction.
Vectors Lesson 1 e (10/17)
The resultants of vectors can be measured using two methods: the parallelogram method and the trigonomic method. Both require drawing diagrams and come out with similar results.Vector Resolution
A vector directed in two dimensions can be thought of as having two components.
The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution that we will examine are
Parallelogram Method of Vector Resolution
The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. If one desires to determine the components as directed along the traditional x- and y-coordinate axes, then the parallelogram is a rectangle with sides that stretch vertically and horizontally.
Step by step breakdown:
EXAMPLE:
Trigonometric Method of Vector Resolution
The trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. Trigonometric functions relate the ratio of the lengths of the sides of a right triangle to the measure of an acute angle within the right triangle. They can be used to determine the length of the sides of a right triangle if an angle measure and the length of one side are known.
Step by step breakdown:
- Construct a rough sketch (no scale needed) of the vector in the indicated direction. Label its magnitude and the angle that it makes with the horizontal.
- Draw a rectangle about the vector such that the vector is the diagonal of the rectangle. Beginning at the tail of the vector, sketch vertical and horizontal lines. Then sketch horizontal and vertical lines at the head of the vector. The sketched lines will meet to form a rectangle.
- Draw the components of the vector. The components are the sides of the rectangle. The tail of each component begins at the tail of the vector and stretches along the axes to the nearest corner of the rectangle. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right)
- Meaningfully label the components of the vectors with symbols to indicate which component represents which side
- To determine the length of the side opposite the indicated angle, use the sine function. Substitute the magnitude of the vector for the length of the hypotenuse. Use some algebra to solve the equation for the length of the side opposite the indicated angle
- Repeat the above step using the cosine function to determine the length of the side adjacent to the indicated angle.
EXAMPLE:Vectors Lesson 1f (10/18)
Component Method of Vector AdditionWhen the two vectors are added head-to-tail as shown below, the resultant is the hypotenuse of a right triangle.
A right triangle has two sides plus a hypotenuse; so the Pythagorean theorem is perfect for adding two right angle vectors.
Addition of Three or More Right Angle Vectors
Example 1:
A student drives his car 6.0 km, North before making a right hand turn and driving 6.0 km to the East. Finally, the student makes a left hand turn and travels another 2.0 km to the north. What is the magnitude of the overall displacement of the student?
The head-to-tail vector addition diagram is shown below.
The order in which the three vectors are added must be changed in order to make a right triangle. If the three vectors are added in the order 6.0 km, N + 2.0 km, N + 6.0 km, E, then the diagram will look like this:
After rearranging the order in which the three vectors are added, the resultant vector is now the hypotenuse of a right triangle. The lengths of the perpendicular sides of the right triangle are 8.0 m, North (6.0 km + 2.0 km) and 6.0 km, East. The magnitude of the resultant vector (R) can be determined using the Pythagorean theorem.
R2
(8.0 km)2 + (6.0 km)2 R2
64.0 km2+ 36.0 km2 R2
100.0 km2 R
SQRT (100.0 km2) R = 10.0 km
Adding vectors A + B + C gives the same resultant as adding vectors B + A + C or even C + B + A. As long as all three vectors are included with their specified magnitude and direction, the resultant will be the same.
The resultant of the addition of three or more right angle vectors can be easily determined using the Pythagorean theorem. Doing so involves the adding of the vectors in a different order.
SOH CAH TOA and the Direction of Vectors
The convention is known as the counter-clockwise from east convention, often abbreviated as the CCW convention. Using this convention, the direction of a vector is often expressed as a counter-clockwise angle of rotation of the vector about its tail from due East.
Example:
6.0 km, N + 6.0 km, E + 2.0 km, N.
In this problem, we know the length of the side opposite theta (Θ) - 6.0 km - and the length of the side adjacent the angle theta (Θ) - 8.0 km. The tangent function will be used to calculate the angle measure of theta (Θ).
Tangent(Θ) = Opposite/Adjacent Tangent(Θ) = 6.0/8.0 Tangent(Θ) = 0.75 Θ = tan-1 (0.75) Θ = 36.869 …° Θ =37°
The result is 37° east of north. Since the angle that the resultant makes with east is the complement of the angle that it makes with north, we could express the direction as 53° CCW.
In summary, the direction of a vector can be determined by finding the angle of rotation counter-clockwise from due east.
Addition of Non-Perpendicular Vectors
It is possible to force two (or more) non-perpendicular vectors to be transformed into other vectors that do form a right triangle. The trick involves the concept of a vector component and the process of vector resolution.
Example:
Max plays middle linebacker for South's football team. During one play in last Friday night's game against New Greer Academy, he made the following movements after the ball was snapped on third down. First, he back-pedaled in the southern direction for 2.6 meters. He then shuffled to his left (west) for a distance of 2.2 meters. Finally, he made a half-turn and ran downfield a distance of 4.8 meters in a direction of 240° counter-clockwise from east (30° W of S) before finally knocking the wind out of New Greer's wide receiver. Determine the magnitude and direction of Max's overall displacement.
As is the usual case, the solution begins with a diagram of the vectors being added.
The resultant is the vector sum of these three vectors; a head-to-tail vector addition diagram reveals that the resultant is directed southwest. Of the three vectors being added, vector C is clearly the nasty vector. Its direction is neither due south nor due west. The solution involves resolving this vector into its components.
Vector C makes a 30° angle with the southern direction. By sketching a right triangle with horizontal and vertical legs and C as the hypotenuse, it becomes possible to determine the components of vector C. This is shown in the diagram below. The side adjacent this 30° angle in the triangle is the vertical side; the vertical side represents the vertical (southward) component of C - Cy. So to determine Cy, the cosine function is used. The side opposite the 30° angle is the horizontal side; the horizontal side represents the horizontal (westward) component of C – C. The cosine function is used to determine the southward component since the southward component is adjacent to the 30° angle. The sine function is used to determine the westward component since the westward component is the side opposite to the 30° angle.
Now our vector addition problem has been transformed from the addition of two nice vectors and one nasty vector into the addition of four nice vectors.
With all vectors oriented along are customary north-south and east-west axes, they can be added head-to-tail in any order to produce a right triangle whose the hypotenuse is the resultant.
The triangle's perpendicular sides have lengths of 4.6 meters and 6.756 meters. The length of the horizontal side (4.6 m) was determined by adding the values of B (2.2 m) and Cx (2.4 m). The length of the vertical side (6.756… m) was determined by adding the values of A (2.6 m) and Cy (4.156… m). The resultant's magnitude (R) can now be determined using the Pythagorean theorem.
R2
(6.756… m)2 + (4.6 m)2 R2
45.655… m2 + 21.16 m2 R2
66.815… m2 R
SQRT(66.815… m2 ) R = 8.174 … m R = ~8.2 m
tangent(Θ) = (6.756… m)/(4.6 m) = 1.46889…
so…
Θ = tan-1 (1.46889…) = 55.7536… ° Θ = ~56°
This 56° SW angle is the angle between the resultant vector . The (CCW) can be determined by adding 180° to the 56°. So the CCW direction is 236°.
Example:
Cameron Per (his friends call him Cam) and Baxter Nature are on a hike. Starting from home base, they make the following movements.
A: 2.65 km, 140° CCW B: 4.77 km, 252° CCW C: 3.18 km, 332° CCW
This is the angle measure diagrams.
A table to organize values.
A
2.65 km
140° CCW
= 2.030… km, West
= 1.703… km, North
B
4.77 km
252° CCW
= 1.474… km, West
= 4.536… km, South
C
3.18 km
332° CCW
= 2.808… km, East
= 1.493… km, South
Sum of
A + B + C
(0.696 km)2 + (4.326 km)2 R2
0.484 km2 + 18.714 km2 R2
19.199 km2 R
SQRT(19.199 km2) R = ~4.38 km
Tangent(Θ) = opposite/adjacent
Tangent(Θ) = (4.326 km)/(0.696 km)
Tangent(Θ) = 6.216
Θ = tan-1(6.216)
Θ = 80.9°
It would be worded as 80.9° SW. Since west is 180° counterclockwise from east, the direction could also be expressed in the counterclockwise (CCW) from east convention as 260.9°.
FINAL RESULTANT: 4.38 km with a direction of 260.9° (CCW).
Vectors Lesson 1 g+h (10/18)
On occasion objects move within a medium that is moving with respect to an observer. In such instances, the magnitude of the velocity of the moving object (whether it be a plane or a motorboat) with respect to the observer on land will not be the same as the speedometer reading of the vehicle. Motion is relative to the observer. The observer on land, often named (or misnamed) the "stationary observer" would measure the speed to be different than that of the person on the boat. The observed speed of the boat must always be described relative to who the observer is. To illustrate this principle, consider a plane flying amidst a tailwind. A tailwind is merely a wind that approaches the plane from behind, thus increasing its resulting velocity. The resultant velocity of the plane (that is, the result of the wind velocity contributing to the velocity due to the plane's motor) is the vector sum of the velocity of the plane and the velocity of the wind. This resultant velocity is quite easily determined if the wind approaches the plane directly from behind.
If the plane encounters a headwind, the resulting velocity will be less than 100 km/hr. Since a headwind is a wind that approaches the plane from the front, such a wind would decrease the plane's resulting velocity. Now consider a plane traveling with a velocity of 100 km/hr, South that encounters a side wind of 25 km/hr, West. The resulting velocity of the plane is the vector sum of the two individual velocities. To determine the resultant velocity, the plane velocity (relative to the air) must be added to the wind velocity. This is the same procedure that was used above for the headwind and the tailwind situations; only now, the resultant is not as easily computed. Since the two vectors to be added - the southward plane velocity and the westward wind velocity - are at right angles to each other, the Pythagorean can be used. In this situation of a side wind, the southward vector can be added to the westward vector using the usual methods of vector addition. The magnitude of the resultant velocity is determined using Pythagorean theorem. The direction of the resulting velocity can be determined using a trigonometric function. Since the plane velocity and the wind velocity form a right triangle when added together in head-to-tail fashion, the angle between the resultant vector and the southward vector can be determined using the sine, cosine, or tangent functions. The tangent function can be used. Like any vector, the resultant's direction is measured as a counterclockwise angle of rotation from due East.
The affect of the wind upon the plane is similar to the affect of the river current upon the motorboat. If a motorboat were to head straight across a river, it would not reach the shore directly across from its starting point. The river current influences the motion of the boat and carries it downstream. Motorboat problems such as these are typically accompanied by three separate questions:
The first of these three questions: the resultant velocity of the boat can be determined using the Pythagorean theorem (magnitude) and a trigonometric function (direction). The second and third of these questions can be answered using the average speed equation.
ave. speed = distance/time. A force vector that is directed upward and rightward has two parts - an upward part and a rightward part. That is to say, if you pull upon an object in an upward and rightward direction, then you are exerting an influence upon the object in two separate directions - an upward direction and a rightward direction. These two parts of the two-dimensional vector are referred to as components. A component describes the affect of a single vector in a given direction. Any force vector that is exerted at an angle to the horizontal can be considered as having two parts or components. The vector sum of these two components is always equal to the force at the given angle. Any vector - whether it is a force vector, displacement vector, velocity vector, etc. - directed at an angle can be thought of as being composed of two perpendicular components. These two components can be represented as legs of a right triangle formed by projecting the vector onto the x- and y-axis.The two perpendicular parts or components of a vector are independent of each other. All vectors can be thought of as having perpendicular components. In fact, any motion that is at an angle to the horizontal or the vertical can be thought of as having two perpendicular motions occurring simultaneously. These perpendicular components of motion occur independently of each other. Any component of motion occurring strictly in the horizontal direction will have no affect upon the motion in the vertical direction. Any alteration in one set of these components will have no affect on the other set.
Vectors Lesson 2 a+b (10/19)
Lesson ACentral idea: A projectile is an object upon which the only force is gravity. Gravity causes a vertical motion, which causes a vertical acceleration.
- What is a projectile?
- It is an object dropped from rest or thrown vertically upwards at an angle to the horizontal
- It is any object projected or dropped that continues in motion by its own inertia
- What forces affect upon a projectile?
- The only force is of downward gravity
- What is inertia?
- The resistance an object has to a change in a state of motion
- How do we represent projectiles?
- A free-body diagram is a good representation of projectiles.
- What type of motion will we see from this diagram?
- We will see a Parabolic Trajectory, similar to a diagram in free fall.
Lesson BCentral idea: A projectile has two components: a vertical motion and horizontal motion. Vertical motion is effected by the downward force of gravity, and therefore is always changing. However, horizontal motion is unaffected by gravity, and therefore stays at a constant rate.
Vectors Lesson 2 c (10/20)
Central Idea: Depending on the projectile, it can either be moving against time solely, or considering the property of displacement.Both of these properties can be addressed numerically.