The Ballistic Pendulum Physics 110 Laboratory Angle indicator Vertical upright θ R cm R b Trigger String cm Projectile Launcher Ballistic Pendulum Base m v cm after h Ramrod Steel ball before In this experiment you will determine the muzzle velocity of a gun using two different methods. The first method consists of firing a ball horizontally from the tabletop by measuring the range of the ball. In the second part of the experiment, the ball will be fired into the ballistic pendulum shown above and the muzzle velocity will be determined by measuring the angle through which the pendulum rises and applying conservation principles. Muzzle Velocity from Projectile Kinematics 1. Consider the situation where a ball is fired horizontally with a velocity v 0 from a height y 0 above the floor and hits the floor a horizontal distance, x f, from the launcher. Derive an expression for v 0 in terms of x f and y 0. 1
2. Clamp the apparatus near one end of the table with the launcher aimed away from the table and remove the pendulum by unscrewing the bearing pin at the pivot. 3. Place the metal ball in the launcher and use the ramrod to cock the piston to medium range. 4. Fire the gun and note where the ball lands. 5. Tape a sheet of white paper to the floor at the point of impact and place a piece of carbon paper on top of it. 6. Fire the gun ten times, making sure that marks are being made on your paper. Measure and record in a data table the distance from the point directly below the release point of the ball to each of your ten marks. 7. Determine the average horizontal distance x f, the ball travels along with the standard deviation of your measurements. We will use this standard deviation as the uncertainty, x f, in your measurement of x f. Record the result as x f ± x f. 8. Measure and record the vertical distance, y 0, the ball falls. Estimate the uncertainty in this measurement, y, and record your measurement as y 0 ± y. 9. Use the expression you derived in part 1 with the average value for x f and the measured value for y 0 to determine the muzzle velocity, v 0. Also determine the uncertainty in v 0 from the uncertainties in x f and y 0 and record the result as v 0 ± v 0. Be sure to include units. Trial 1 2 3 4 5 6 7 8 9 10 Avg. σ x f (m) x f : ± y 0 : ± v 0 : ± 2
Determining Muzzle Velocity Using the Ballistic Pendulum If we look at the situation right after the ball impacts the pendulum, we see that the energy is purely kinetic, E 0 = 1 2 (M +m)v2, where M is the mass of the pendulum, m is the mass of the ball, and v is the speed of the center of mass of the combined system after the collision. The pendulum now swings up to an angle θ and stops, so that all energy has been converted to gravitational potential energy, E f = (M + m)g h where h is the change in height of the center of mass of the ball/pendulum system. From the figure on Page 1, it s clear that h = R cm (1 cos θ). To determine v, we use the conservation of momentum and neglect the force exerted by the pivot during the collision: mv 0 = (M + m)v. We would like to have an expression for v 0 in terms of only measured quantities, so squaring this ergo m 2 v 2 0 = 2(M + m) 1 2 (M + m)v2 = 2(M + m)e 0 = 2(M + m) 2 gr cm (1 cos θ), v 0 = M + m 2gRcm (1 cos θ). m 10. Load and cock the launcher to medium range. Return the pendulum to its vertical position and move the angle indicator to zero degrees. 11. Fire the launcher and note the angle reached. 12. Load the launcher again, then set the angle indicator to 5 less than that reached in step 11. This will nearly eliminate the drag on the pendulum caused by the indicator, since the pendulum will only move the indicator the last few degrees. 13. Fire the launcher and record the angle reached by the pendulum. Repeat and record the angle each time to get a total of ten angle measurements. 14. Determine the average angle. Also, calculate the standard deviation of your measurements and use this as the uncertainty, θ, in your measurement of θ. Record the result as θ ± θ. Trial 1 2 3 4 5 6 7 8 9 10 Avg. σ θ (degrees) θ: ± M: ± m: ± 3
15. Remove the pendulum from the base and measure the mass of the pendulum. Estimate the uncertainty and record the result as M ± M. 16. Measure the mass of the ball, estimate the uncertainty, and record the result as m ± m. 17. Determine the center of mass of the pendulum with the ball by balancing it on the edge of a ruler. Measure the distance from the pivot point to this balance point, estimate the uncertainty, and record it as R cm ± R cm. R cm : ± 18. Use the expression derived above with the data you measured in this section to calculate the muzzle velocity of the launcher. Also, determine the uncertainty in v 0 from the uncertainties of your measurements and record the result as v 0 ± v 0. v 0 : ± 19. Do the values for the muzzle velocity determined using the two different methods agree to within experimental uncertainties? Calculate a percent difference. 20. What sources of error are there in this experiment? How much do these errors affect your results? 4
21. What percentage of the kinetic energy is lost in the collision between the ball and the pendulum? Would it be valid to assume that kinetic energy was conserved in the collision? 22. How does the angle reached by the pendulum change if the ball is not caught by the pendulum? You can test this by turning the pendulum around so that the ball strikes the back of the pendulum instead of the catcher. Is there more or less mechanical energy transferred to the pendulum? 5
23. (optional) The error estimate calculated by the extreme values for measured values isn t quite right. A multivariate function v 0 = v 0 (M, m, R cm, θ) that has variance in all its parameters has variance v 0 = [( ) 2 v0 M] + M [( ) 2 [( ) ] 2 v0 v0 m] + R cm + m R cm where θ must be in radians. The expressions for the derivatives are v 0 M = 2gRcm (1 cos θ) m v 0 m = M 2gRcm (1 cos θ) m 2 v 0 (M + m) 2g(1 cos θ) = R cm 2m R cm v 0 θ = (M + m)gr cm sin θ m 2gR cm (1 cos θ) [( ) ] 2 v0 θ θ Using Excel, and your data, find the uncertainty in v 0 with this error propagation. 6