Projectile Motion

Purpose

In this exercise we will use the same VideoPoint analysis methods we used with a simple falling object to look at motion in two dimensions. We will see the independence of the horizontal and vertical components of the motion, and use this to be able to predict the distance a projectile will travel, given its initial velocity and the angle of launch.

Theory

You have learned that under the influence of gravity alone, objects accelerate downwards only. Not only do they accelerate downward but, regardless of the mass of the object, it will accelerate at a constant 9.8 m/s² downward. However, since gravity only acts downward, the horizontal velocity will remain constant in the absence of other forces (e.g., air resistance). The horizontal and vertical components of motion for freely falling bodies are completely independent, so we can calculate the two separately.

We have the same basic three equations governing the motion, except now we have these three equations for both the x direction and the y direction. We add one further complication in projectile motion, that the object’s velocity at the beginning is not necessarily zero. We’ll assume, though, that the starting position is zero. So, our equations become:

Position:

Velocity:

Acceleration:

To remind us that the acceleration due to gravity is directed downward, we’ve treated g as a positive constant, but explicitly show it subtracted from the initial upward velocity of the projectile.

To determine how long a projectile stays in the air, we can calculate what t is when y value is zero. We can also then use that to determine the distance the object will travel, by calculating the value of x for this value of t.

Experimental Procedure

1. Open the VideoPoint software, and choose one of the two movies showing a ball being thrown. We’ll look at both; one shows a high trajectory, the other shows a lower trajectory.

2. Specify that you will be locating a single object.

3. The first frame of the movie will appear, and the mouse pointer will be a target. Click on the steel ball to mark its position in this frame. The movie will advance to the next frame; continue marking the ball’s position in each frame. You can use the frame advance and back buttons to move one frame at a time to help spot the ball in frames where it is difficult to see.

6. Again, do not set the position in the frames after the ball bounces at the bottom (and you may also want to ignore the first point before the ball actually starts moving). It will be harder to use the automatic graph fitting capabilities of VideoPoint if you have the extra data points.

7. To provide a scale factor, we can use David’s height! He says he’s about 5’10”, which is 1.78 meters. Click on the ruler icon in VideoPoint, enter the height, and then mark the endpoints as the top of his head and bottom of his feet.

8. You may also want to adjust the origin to make your calculations simpler. You can simply drag the origin to a new position; the best choice is probably the starting point of the ball in the frame where it is released.

9. Choosing New Graph from the View menu will allow you to look at various views of the data. First, leave the horizontal axis as time, and set the vertical axis to display the y-component and choose “Position”. This will just show a chart of the object’s motion. Also look at the velocity and acceleration. These should look very similar to the graphs from the gravity drop experiment, except that we start with an initial velocity greater than zero.

10. Try viewing a graph of the x-component of position. Also look at the velocity and acceleration. You should see that acceleration in this direction is zero, and the velocity remains constant.

11. Fit a curve (use polynomial of degree 2 as the type) to the y position graph. The curve will have an equation of the form At² + Bt + C, where A, B, and C are constant values. Since position depends on t², we want the value of the first constant, A. Looking at the above equations of motion, we see that g should be equal to twice this value.

12. Fit a line (use linear as the type) to the y velocity graph. The line will have an equation of the form At + B, where A and B are constants. Again, looking at the equations governing the motion, the velocty depends directly on t, so the slope of the line (the value of A) will give you the acceleration.

13. Fit a line to the y acceleration graph. The intercept of this line (the value of B) will give you the acceleration (the slope should be zero if acceleration is in fact constant). The line may not be quite flat, so instead of taking the intercept, you may want to determine the value of the fitted line for some value of t near the middle of the fall.

High Arc Acceleration (y direction)

From Position Graph:

From Velocity Graph:

From Acceleration Graph:

14. Fit a line (use linear as the type) to the x position graph. The line will have an equation of the form At + B, where A and B are constants. Again, looking at the equations governing the motion, the position depends directly on t, so the slope of the line (the value of A) will give you the velocity .

15. Fit a line to the x velocity graph. The intercept of this line (the value of B) will give you the velocity (the slope should be zero if acceleration is in fact zero). The line may not be quite flat, so instead of taking the intercept, you may want to determine the value of the fitted line for some value of t near the middle of the fall.

High Arc Velocity (x direction)

From Position Graph:

From Velocity Graph:

16. Save your video analysis.

17. Repeat this procedure for the other movie (the low arc throw if you started with the high arc throw, or vice versa).

Low Arc Acceleration (y direction)

From Position Graph:

From Velocity Graph:

From Acceleration Graph:

Low Arc Velocity (x direction)

From Position Graph:

From Velocity Graph:

Analysis

You can determine the total initial velocity, as well as the angle of the throw, by combining both the x and y values for velocity. Your text has an excellent discussion of projectile motion beginning on page 118. Some simple results are shown below. For the total velocity, we use vector addition, which is just the Pythagorean theorem:

To determine the angle the projectile was launched at, we take the ratio of the vertical velocity to the horizontal velocity, and then take the inverse tangent.

Calculate the total velocity and launch angle of both the high arc and low arc trajectories. You can use whichever method for determining the x and y components of the velocity you wish.

High Arc Total Velocity:

High Arc Launch Angle:

Low Arc Total Velocity:

Low Arc Launch Angle:

As noted in the text on page 122, you can calculate the component velocities from the total velocity and the angle. Once you’ve done this, you can determine the time the object will take to return to the ground based on the initial upward velocity. Then, using that value of t, you can multiply by the horizontal velocity to determine the range.

Given a total velocity equal to the one you calculated for the high arc trajectory, calculate the range for several different angles.

v0	θ	vy0 = v0 sinθ	vx0 = v0 cosθ	tf = 2 vy0/g	xf = vx0tf
	15°
	30°
	45°
	60°
	75°
	90°

Questions

Did the vertical acceleration stay constant at about g?

Did the horizontal acceleration stay constant at about 0?

Was the vertical velocity a line with slope g?

Was the horizontal velocity a flat line?

Is the graph of the vertical distance covered a parabola?

Is the graph of the horizontal distance covered nearly a straight line?

If your value does not exactly coincide with the expected value, what effects might have caused this?

How accurately do you think that you measured the force of gravity?