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Introductory Mechanics Experimental Laboratory
Projectile Motion
Goals: Observe motion in two dimensions. Use simple measuring devices to make approximations. Understand the limits of precision in measurement.
APPARATUS This experiment uses two glass tubes to launch a ball horizontally. Both tubes use grav- ity to get the ball rolling, and a straight section to launch the ball. A long tube will be used in part A to find the range of a ball. A short tube will be used in part B to plot the trajectory of the ball.
In part A, the end of the long tube is situated at a height ( h ) above the ground. The ball comes out of the tube with an initial horizontal velocity ( v 0 ). As the ball flies through the air it will curve downward due to gravity and strike the floor some distance ( d ) away from the end of tube.
In part B the ball curves downward and its path can be measured with a vertical board sided with carbon paper. The board can be place at various distances ( x ) away from the end of the tube. The impact of the ball on carbon paper leaves a mark that can be mea- sured at the end of the experiment. The sequence of carbon paper marks repesents the vertical position ( y ) of the ball at different points in flight.
THEORY The ball comes out of either tube in the direction of the last section of the tube. This direction is purely in the horizontal direction at some initial velocity v 0. This velocity can be measured by finding the time T that it takes to go between two points in the hori- zontal tube separetd by a distance ( D ). The velocity is then the
(EQ 1)
A projectile in the air is subject to an acceleration ( a ) downward due to Earth’s gravity. The acceleration only acts downward and has no effect on the horizontal motion. Since there is no initial vertical motion for the ball, the distance downward is given by
v 0 = D ⁄ T
2 Projectile Motion
(EQ 2)
where h is the initial height and g is the acceleration due to gravity, g = 9.8 m/s^2.
The time it takes to hit the ground is found by setting y equal to 0 (the ground) and solv- ing EQ 2 for t.
(EQ 3)
There is no acceleration in the horizontal direction so the forward distance covered when the ball hits the ground, called the range ( R ), is found by substituting the time from EQ 3 into an expression for distance: d = v 0 t , or
(EQ 4)
A similar set of equations applies to the motion of a projectile before it reaches the ground. The vertical distance down from the starting point is y = gt^2 and the horizontal distance is x = v 0 t. If these two equations are combined to eliminate t , then we get
(EQ 5)
This equation that relates two distances describes the trajectory of the projectile.
DATA COLLECTION Part A - Long glass tube
1. Measure the height of the long tube opening above the floor. This is the vertical dis- tance ( h ) that the ball will fall. 2. Use a timer to determine how long it takes the ball to travel the distance between two marks separated by one meter while in the tube. Repeat your timing five times, recording each time, and take an average to use as time ( T ). 3. Calculate the horizontal speed of the ball ( v 0 ) in the long level portion of the tube with D = 1 m using EQ 1. 4. Use the height ( h ) and horizontal speed ( v 0 ) to predict how far out from the spot on the floor directly beneath the opening of the tube the ball will land using EQ 4. This is the range ( R ). 5. Test your prediction by taping some carbon paper to the floor where you expect the ball to hit. Send the ball through the tube ten times letting it hit the paper each time to make a mark on the carbon paper. 6. Measure the horizontal distance from the spot on the floor directly beneath the open- ing to the edge of the paper. 7. Measure and record the distance of each mark on the paper from the edge of the paper and find an average.
y y 0 1 2
- --- at^2 h^1 2 = = –--- gt^2
t^2 h g
= ------
R v 0 2 h g
= ------
y g x v 0 -----^
2 h g v 02
= = –------- x^2
