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Trajectories

When a projectile is fired with a given initial velocity, as determined by its initial speed and angle of elevation, the trajectory can be predicted quite well. For high velocities there is resistance from the air which causes a deceleration or drag on the object in the horizontal direction. For low speeds this can be safely ignored with little noticeable effects. Consider the system below:

Figure 1: Trajectory Apparatus. Horizontal and Vertical components of flight are analyzed separately.

The basic equations of motion as derived from elementary principles are:
 
a = constant (1)
v = vo + at (2)
d = vot + 1/2at2 (3)

where a is acceleration, v the instantaneous velocity, vo is the initial velocity, t is the time of flight and d the distance traveled in a particular dimension. By breaking the flight of the projectile into its x- and y- components we have:
 

x y
1 ax = 0 ay = -g
2 vx = vox vy = voy - gt
3 x = vox t y = voy t - 1/2gt2

Table 1: Vector Components of Trajectory Variables.

If the horizontal distance of flight and the height of the projectile at the start of motion (x and y distances) are known, essentially all of the variables may be calculated for the flight. The entire trajectory is then known.

Your mission, should you decide to accept it, is to determine the initial velocity of the projectile, and the time of flight.

Measure the increment of horizontal distance traveled in the increment of time. This should be done several times so that an average can be calculated by standard methods. Find an average horizontal velocity and the initial velocity can be calculated from the cosine relationship. Then time can be easily determined. This technique will not work by measuring the velocity in the vertical direction since it is accelerated in that direction and a complex iterative technique would need to be employed.

OBJECTIVES:

  1. To determine the initial velocity of the projectile and the time of flight.
  2. To study the parabolic trajectory of a projectile.
  3. To usedigital video analysis techniques to record time dependent data.

APPARATUS:

PROCEDURE:

          One tool that is available for you today is the use of digital video. This is a powerful tool. Camcorders typically record frames 30 times a second. So the time between frames is 1/30 th second. That's cool.  Here's how to proceed:

1. Select the icon from your desktop called VideoPoint and double click on it.

2. The Video Point screen will come up. It has some dumb  useless  information so click it away. When you do so you should get a menu that will let you OPEN MOVIE. Click on OPEN MOVIE. Your instructor will make sure you have the CD in your computer. Select d:// movies/PASCO/104.mov    Again, your instructor will help. Now you can acquire data for an object in some kind of accelerated motion. Admittedly there is friction involved so it is not exactly frictionless, but at low velocities friction is small.

3. Click on the ball in successive steps as it moves across the screen. Be sure you calibrate the background so you can convert pixels to cm or metres. The computer sets clicks of the cursor so you have x and y data points.

4. Plot in EXCEL the displacement and velocity versus time.; ie, x vs t and y vs t and x vs y.  Remember every time frame is 1/30th second. Velocity is merely the change of position (take the current position less the previous value) to get velocities.

5. Compare the velocities in the x-direction. Is it uniform? What does that say about friction in the x-direction?

6. Do this for the y-direction. The velocity is accelerated. What can you say here?

7. Plot x vs y. This should be a parabolic trajectory.

6. Alternatively you can make your own video of a dropping golf ball or something like that. Ask your instructor
for help.

DATA ANALYSIS:

  1. Calculate Vo and t with appropriate uncertainties. Do this by measuring the horizontal increments and then determining the x-component of velocity, Vx. The x-component is uniform since at low velocities air resistance is negligible. Report your results.
  2. It may be best to estimate uncertainties rather than calculating them. Basically, can you reliably estimate positions on the video screen to within a centimetre? If so, this may make more sense than calculating the effect of + one or two pixels and carrying through the uncertainties through pages of calculations. Justify your determinations to the reader.
  3. Plot the trajectory on linear graph using Excel. Use the distance horizontal vs. distance vertical for your axes.
  4. Discuss the results and report the uncertainties as well as what else you might predict about the trajectory in the Summary.
  5. Have fun!
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