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projectile motion

OBJECTIVE

To determine the initial velocity of a projectile by a measurement of its range and compare it to the velocity measured by timing the ball crossing a known distance. The dependence of the range on the firing angle is explored.

EQUIPMENT

Caliper, ballistic pendulum, angle apparatus, clamp, timer w/ gate, carbon paper.

INTRODUCTION

When an object is fired at some angle above the horizontal in a gravitational field, it follows a curved path. If there is little or no air resistance, that path is a parabola. This motion is due to the fact that in the horizontal direction the object experiences no acceleration, while in the vertical direction the object is accelerated by gravity. Using basic kinematics and trigonometry one can derive the so-called range equation.

R=(v2/g)*sin(2θ)

R represents the horizontal distance the projectile travels, v is the magnitude of the initial velocity, g is the acceleration due to gravity and θ is the firing angle. You will attempt to verify this formula using a modified ballistic pendulum apparatus and a photogate.

PROCEDURE

1. Set the apparatus on the floor allowing enough room to avoid hitting anyone with the speeding ball. Set the timing gate at a right angle to the path of the ball to insure proper timing. To assure you have the gate placed properly, align the infrared beam with the metal dowel on the gun. Ask your lab instructor if you need help. Place the hole in the ball over the metal dowel of the gun and push the ball onto the spring mechanism until you reach the last position. (You may need to depress the trigger slightly while you push the plunger into the last position.) 

2. The timer should be set on "Time", Stopwatch". Press the start/stop to display a *. The timer will display the duration the ball blocked the sensor in the timing gate. 

3. Before you fire the gun depress the red reset button on the timer. Test the gun by depressing the trigger on the top of the spring mechanism with an even pressure. Trying to reproduce this motion each time you fire the gun will give you more consistent results. Tape several pieces of paper to the floor in the general location of the test hit and place sheets of carbon paper on it. This will allow you to measure the distance from the gun to the hit spot with the carbon paper marking the spot. Accurately measure of the width of the ball. This is used with the time displayed on the timer to find the initial velocity of the ball. 

4. Measure the range for firing the projectile at angles of 5°, 15°, 30°, 45°, 60°, and 70° and the time the ball blocked the timer gate. You should launch the ball from each angle at least 5 times. It is imperative that you "cock" the gun to the same position each trial. If you do not do this, you will not get consistent results.

ERROR ANALYSIS

5. Examine your data. All time measurements should be very close to one another. If they are not you may have to repeat one or more of the firing angles. Determine the initial speed for each time. Average these speeds and find the greatest deviation from the average. This deviation is you uncertainty in initial speed, Δv.

6. You must estimate the uncertainty in the firing angle. This will probably vary from one angle to the next, and be fairly large for angles above 45°. Within your group, decide how you will determine this uncertainty, Δθ, and record it. Also, record your method for determining Δθ.

7. To determine the uncertainty in the predicted range for each firing angle, use the following formula.

ΔR = (((2V/g)ΔVsin(2θ))2 + ((V2/g)Δθ2cos(2θ))2)1/2

In the above, both θ and Δθ must be expressed in radians.

8. Finally, determine the shot spread from your experiment. You will have to decide how large a box is needed to enclose all hits.

GRAPHS AND DIAGRAMS

1a) Plot your experimental data for the range of the ball versus the sine of twice the firing angle (sin(2()). 

1b) Calculate the theoretical range as a function of sine of twice the firing angle and plot a curve representing this on the same graph with your experimental data. 

2) Make a free body diagram for the projectile for the situation when it is in flight. Indicate which forces are constant (if any) and which change with time (if any). 

QUESTIONS AND CALCULATIONS

1) Compare your measured initial velocity from the timer to that calculated from the range and firing angle data. 

2) What conclusions can you draw from graph 1? How does the curve compare to the measured data points: is it a good representation of the data for all firing angles or is it better at small angles or large angles? 

3) Compare the shot spread determined in part 8 with the uncertainty in the range from part 7. How does the uncertainty in the range affect your comparison to the calculated range?

3) Is mechanical energy conserved in this experiment or is some lost? On what do you base this claim? 

4) At what angle should you fire the gun to get the greatest range? Do your data support this claim? Explain. 

CAVEATS

1. You must be certain to keep the gun apparatus in one location throughout the experiment. Failure to do so will affect the range.

2. The range equation is only valid if the projectile lands at the same height from which it is fired. Develop some way to take this into account.

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