Ballistic Pendulum and Projectile Motion The initial velocity of a ball shot from a spring gun is determined by the equations for projectile motion and by the equations for a ballistic pendulum. Projectile motion: For a projectile whose initial velocity is horizontal. The initial velocity (v i ) can be calculated from the measurement of the range x and the vertical distance y as shown in Fig. 1. Fig. 1 In such a case it must be remembered that the horizontal component of the velocity always remains constant and is not affected by the constant downward pull of gravity. Furthermore the vertical component of the motion is unaffected by its horizontal flight and hence the body falls vertically, as does a freely falling object having no horizontal motion. The actual motion of such a projectile is the combination of its horizontal constant velocity and its downward uniformly accelerated velocity. During the time interval t required for the projectile to reach the floor, it will have moved horizontally through a distance v ix = x t (l) and during the same interval, because of the acceleration due to gravity g, it will have fallen a distance y = 1/2 gt 2. (2) The time of the motion can be found from equation 2. Then, the initial velocity can be found. Ballistic Pendulum: For a system where there is no external force, momentum of the system is conserved. That is, the momentum before the collision must be equal to the momentum after the collision. In the present case, an inelastic collision occurs between the ball from the spring gun and the pendulum. 42
The momentum of the ball just before impact shall be equal to the combined momenta of the ball and pendulum an instant after impact. In the ballistic pendulum used in this experiment the velocity of the pendulum before impact is zero, and hence its momentum before impact is zero. The momentum of the ball before impact is the product of its mass m and its initial velocity v i just before impact. Since the projectile becomes imbedded in the pendulum after impact, the ball and pendulum an instant after impact have a common velocity v f and the combined momenta is (m b )v f. From the law of the conservation of momentum Total momentum before impact=total momentum after impact m b v i = (m b )v f (3) from which the initial velocity is given by v f = m b v i (m b ) (4) As a result of the impact, the pendulum containing the projectile swings about its point of support, and thus the center of gravity rises through a vertical distance h. Knowing this velocity after impact, it is possible to determine the height. The kinetic energy of the system an instant after impact must, by conservation of energy, equal the increase in potential energy gained by the pendulum when it reaches its highest point. By equating the kinetic energy to the potential energy: KE = PE g 1 2 (m + m )v 2 b p f = (m b )gh from which h = v2 2g (5) (6) If the maximum height the center of mass the pendulum swings through is h and the length of pendulum from the center of mass to the point of rotation is L, then the angle θ is given by " θ = cos 1 1 h % $ ' # L & Compare this angle to the average angle obtained from your experiemtal trials and calculate a percent error. (7) 43
Apparatus: 1. Ballistic pendulum 2. Scale 3. Tape measure 4. Steel Marble 5. Plumb bob 6. Carbon paper PROCEDURE 1. In the first part of the experiment, the initial velocity of the projectile is obtained from measurements of the range and fall. The apparatus should be set near one edge of a level table. In this part the pendulum is not used and should be swung up onto the rack so that it will not interfere with the free flight of the ball. 2. Get the gun ready for firing by placing the ball inside the spring gun and using the plunger, push the ball in the gun until the trigger is engaged at the highest setting. Make sure the ball is resting against the spring and has not rolled toward the front of the gun. The ball is fired horizontally so that it strikes a target placed on the floor. Fire the ball and determine approximately where it strikes the floor. Place a sheet of white paper on the floor so that the ball will hit it near its center and cover it with carbon paper; tape the corners of the paper to keep it from moving around. In this way a record can be obtained of the exact spot where the ball strikes the floor. Fire the ball a total of six times. 3. Measure the horizontal distance from the point of projection (There is a circle with an x inside on the side of the spring gun.) to the point in the middle of the cluster of shots with the floor. Use a plumb bob to locate the point on the floor directly below the ball as it leaves the gun. 4. Measure the vertical fall of the ball, that is, the vertical distance of the point of projection above the floor. 5. Calculate the initial velocity of the ball by using equations (1) and (2). 6. Calculate the final velocity of the ball and the pendulum using conservation of momentum (4). Show work on the data sheet and record the results in the table. 7. Calculate the height of the pendulum using equation (6) and the predicted angle of the pendulum using equation (7). 8. Loosen the thumbscrew at the top of the pendulum and carefully remove the pendulum from its support. Weigh the pendulum and the ball separately and record the values obtained. Measure the length of the pendulum from the axis of rotation to the center of mass of the pendulum. 9. Get the gun ready for firing. Release the pendulum and allow it to hang freely. Reset the angle indicator to zero. When the pendulum is at rest, pull the trigger, thereby firing the ball into the pendulum. This will cause the pendulum with the ball inside it to swing up and drag the angle indicator with it. Record the maximum angle on the data sheet. Repeat five times. 10. Compute the average angle of the pendulum. And compare to the calculated angle. Find a percent error. 44
Projectile Motion: Data Horizontal distance of the Projectile (x) = Vertical distance (h) = Show work for initial velocity calculation below: Initial velocity of the ball by projectile motion (v i )= Mass of ball (m b ) = Mass of Pendulum (m p ) = Show work for final velocity calculation below: Final velocity of the ball and pendulum (v f )= Show work for height of pendulum below: Height of pendulum (h) = Length of Pendulum (L) = Show work for angle of pendulum below: Calculated angle of pendulum (θ) = Experimental Angle (θ) = 1. 2. 3. 4. 5. Average experimental angle (θ) = Percent difference between calculated and experimental θ = 43
QUESTIONS 1. For the ballistic pendulum, calculate the kinetic energy of the ball before the collision and the kinetic energy of the ball and pendulum just after the collision. 2. Using the result of question 1, find the fraction of the kinetic energy lost during the collision with the initial kinetic energy. Compare this fraction to the ratio of the mass of the pendulum to the total mass of the pendulum and the ball. Fraction Lost = KE f KE i KE i = m p m b 42