Purpose:

To reinforce how to calculate linear displacement, linear velocity, and linear acceleration, as well as learning how to find the horizontal and vertical components of velocity and acceleration.  In addition, you will learn about the equations for uniform acceleration which can be used to calculate the height and distance of a projectile.

Videotaping:

• Step 1: One person should stand with their throwing arm side facing the video camera (sagittal view). Adjust the video camera by zooming in or out until you can see the upper body and arm when it is extended over the head, plus some spare room to see the ball as it is thrown. You should make the picture as large as possible while still leaving room to see the arm throughout the throwing motion and the ball as it moves away after the throw.
• Step 2: Once you set the zoom on the video camera do not change it. Videotape a meter stick for 5-10 seconds.
• Step 3: Videotape a ball being thrown by a skilled and unskilled thrower. Videotape the activity using a shutter speed of 1/500 sec or higher.  This will eliminate a blurry picture.  Measure the distance from the point of the throw to where the ball lands. Also, with a stopwatch, time the ball while it is in the air.

Calculations:

• Step 1: Tape a transparency to the television screen. Using the VCR and TV, advance the videotape to the recording of the meter stick. With permanent markers, mark both ends of the meter stick.
• Step 2: View the trials of the balls being thrown. Choose the one trial for each thrower where the object is least blurry. (Most likely a darker screen and faster shutter speed.)
• Step 3: Push the jog shuttle button on the remote control. Use the outside dial of the jog shuttle control to forward the videotape to the frame where the ball leave the throwing hand. Mark the ball at this frame, placing a star above it, and also mark the toe of the foot farthest in the throwing direction.  Rewind the videotape 3 frames.
• Step 4: Place your finger in the indented circle on the jog/shuttle control and go forwards (clockwise).  Using a permanent marker plot the balls after each frame (one click equals one frame). Continue marking the balls until they go off of the screen.
• Step 5: Remove the transparency and gather a ruler, scientific calculator, and pencil to fill in the table with results. First, measure the distance between the ends of the meter stick in millimeters. You will use this measurement to determine the appropriate scale (convert from video distance to reality distance).
  • For example: meter stick (1m in reality) = 85mm (in video distance)
• Step 6: Measure the distance between each ball in millimeters and convert the measurement into meters using the calculation below. This is your displacement.

  • Linear Displacement: d=p₂-p₁

  • Distance between balls (m) = distance between balls (mm)/meter stick measurement (mm)
  • For example: 22mm/85mm = 0.259m
• Step 7: From each ball mark, draw a line parallel to the bottom/top of the transparency.  Using a protractor measure the angle between this line and the line created by connecting two consecutive ball marks.   This is your trajectory angle.  Record the angle in your table.
• Step 8: The video camera records 60 frames per second or one frame each 0.0167 seconds. Using distance in meters and time in seconds calculate the velocity and acceleration of the balls in each frame. The resultant linear velocity and resultant linear acceleration will be the hypotenuse of each triangle. 

  • Linear Velocity:  v_r=d/Dt

  • Linear Acceleration: a_r=v₂-v₁/Dt

• Step 9: Figure the values for vertical and horizontal velocity and acceleration.

  • Vertical Velocity: v_v=v_r x sinq

  • Horizontal Velocity: v_h=v_r x cosq

  • Vertical Acceleration: a_v=a_r x sinq

  • Horizontal Acceleration: a_h=a_r x cosq

    • Note: In this specific example we cannot determine the velocity in the first frame.  We know that the balls were moving prior to the first frame, but we do not know how fast. The same applies to the acceleration of the balls.

• Step 10: Using the equations for constant acceleration, determine the ball’s maximum height, maximum distance and time airborne.

  • Maximum height 0=(v_v1)²+2a_gd_v
  • Time airborne 0=v_v1+a_gt_a->b (gives time until peak)
  • d_v=(0.5)a_gt_b->d² (gives time from peak to land)
  • Maximum distance d_h=v_h1t_a->d

Questions:

Using Microsoft Excel, make one table and seven graphs, one each for linear displacement, linear velocity, horizontal velocity, vertical velocity, linear acceleration, horizontal acceleration, and vertical acceleration.  Include the data for both balls on the same graph.

1. Graph the resultant velocity. What pattern does it follow? How do your values differ from what you expected?
2. Graph the horizontal velocity. What pattern does it follow? How do your values differ from what you expected?
3. Graph the vertical velocity. What pattern does it follow? How do your values differ from what you expected?
4. Graph the resultant acceleration. What pattern does it follow? How do your values differ from what you expected?
5. Graph the horizontal acceleration. What pattern does it follow? How do your values differ from what you expected?
6. Graph the vertical acceleration. What pattern does it follow? How do your values differ from what you expected?
7. What external force(s) is/are acting on the horizontal velocity?
8. What external force(s) is/are acting on the vertical velocity?
9. What are some possible explanations for the error between what you expected and what you observed?
10. List the actual measured distance and the forumla calculated distance.  How well do they compare?
11. List the actual measured time and the forumla calculated time.  How well do they compare?

horizontal-air resistance
vertical-air resistance, gravity
movement of videocamera
misplaced points-movement of transparency
measurement error-angular or linear displacement
rounding error, unless computed continuously on calculator or MS Excel
videocamera not perpendicular to activity