Experiment 3: Projectile Motion

The aim of this experiment is to measure height using a ruler and calculate the same height using equations of projectile motion and compare. The equipment needed include A photo gate, a computer, a stand, a plane wood, a spherical metal ball, vernier caliper, a meter stick, a thin metal rod, a string and a small weight.

  1. photo gate
  2. computer
  3. stand
  4. plane wood
  5. spherical metal ball
  6. vernier caliper
  7. meter stick
  8. thin metal rod
  9. string
  10. small weight

Theory

A Photo Gate

A photo gate is essentially a stop watch which is triggered by the blocking or unblocking of a light signal that travels from one of its sides to the other side. When in the mode for measuring the speed of small objects that cross it, the stop watch is started when the light signal is blocked and stopped when the light signal is unblocked. Now suppose an object of size d is crossing the photo gate with a speed v. The photo gate will measure the time taken for the size d of the object to cross it. So, if the time measured is t, then the speed may be approximated by

v = d ⁄ t

Even though v is the average velocity over the time interval t, if the object is small, d will be small and v will be a good approximation of the instantaneous velocity of the object as it crosses the photo gate.

Projectile Motion

Projectile motion is motion under gravity. The acceleration due to gravity is directed perpendicularly downward and has a value of -9.8 m ⁄ s 2. Since it doesn’t have a horizontal component, the component of the motion in the horizontal direction is motion with a constant speed while the vertical component of the motion is a uniformly accelerated motion. If an object starts with an initial velocity whose magnitude is vi and makes an angle θi with the positive x-axis, is displaced horizontally by Δx and vertically by Δy; and ends up with a final velocity whose magnitude is vf and makes an angle θf with the positive x-axis in a time interval t, then the motion is governed by the following equations:

vf cos ( θf ) = vi cos ( θi )

Δx = vi cos ( θi ) t

vf sin ( θf ) = vi sin ( θi ) + gt

Δy = vi sin( θi ) + gt 2

{ vf sin ( θf ) } 2 = { vi sin ( θi ) } 2 + 2gΔy

Δy = { vi sin ( θi ) + vf sin ( θf ) } ( t ⁄ 2 )

where the gravitational acceleration g = -9.8 m ⁄ s 2. Now substituting for t from the second equation into the fourth equation of the above equations the following equation can be obtained,

Δy = Δx tan ( θi ) + ( g ⁄ 2) { Δx ⁄ ( vi cos ( θi ) } 2

Therefore, if we know the initial velocity (both magnitude and direction) and the horizontal displacement, we can determine the vertical displacement.

Procedure

  1. Measuring the height of the table using a meter stick

    1. Put the plane wood on the surface of the table flat so that one of its edge is parallel to the edge of the table.
    2. Suspend a weight attached to a string from the edge of the plane wood to determine the point perpendicularly below the edge of the plane wood at the floor.
    3. Measure the height,h, from the point determined in procedue (2) and the edge of the plane wood by means of a ruler.

      measured height ( h ) = m

  2. Determining the height of the table using equations of projectile motion

    1. With the arrangement of the plane wood as in procedure I, set the photo gate right at the edge of the plane wood and connect it to the computer.( select precision timer, photo gates and then any gate)
    2. Measure the diameter ( d ) of the spherical metal ball.

      diameter ( d ) = m = 1.9 m

    3. Now push the metal ball with some speed (the greater the speed the better) in a direction as perpendicular as possible to the edge of the plane wood. Read the time ( t ) taken for the ball to cross the photo gate on the computer.

      time ( t ) = s = 0.08 s

      Record this on the first row of the first column of Table 1.

    4. Locate the point, the ball fell on the ground and measure the horizontal displacement ( Δx ) between this point and the point right below the edge of the plane wood. (You have already determined this point in procedure I.(2)).

      horizontal displacement ( Δx ) = m

      Record this on the first row of the second column of Table 1.

    5. Calculate the initial speed ( vi ) of the ball by dividing its diameter ( d ) obtained in procedure (2) by the time ( t ) taken to cross the photo gate obtained in procedure II. (3).

      initial speed ( vi ) = m ⁄ s

      Record this on the third column of the first row of table 1.

    6. Determine the angle ( θi ) the initial velocity (the velocity with which it is thrown) makes with the horizontal as the ball leaves the plane wood.

      angle ( θi ) = deg = 0 deg

    7. Obtain the calculated height ( h ) from the floor to the edge of the plane wood according to the following equation obtained from equations of projectile motion. (The vertical displacement, Δy, is negative since the displacement is downwards. So, the height ( h ) is obtained as the negative of Δy. ( h = -Δy ).

      calculated height ( h ) = - Δy = - Δx tan ( θi ) + ( |g| ⁄ 2 ) { Δx ⁄ [ vi cos ( θi ) ] } 2 = m

      Record this on the first row of the fourth column of Table 1.

    8. To reduce the error statistically, repeat procedures (3.) through (7) five times to fill Table 1.

      Table 1
      update table

    9. Calculate the average of the heights of column four of table 1.

      average height = m

    10. Compare the measured height obtained in procedure I.3. with the calculated height obtained in procedure II.9. by calculating the percentage error.

      % error = | { measured height (I.3.) - calculated height (II.9.) } ⁄ ( measured height ) | * 100% =