DiscussionTrajectoryThe initial velocity of a projectile can be obtained by firing a ball horizontally and allowing it to fall freely to the ground. The velocity can then be found using equation one for the vertical motion and equation four for the horizontal motion. The initial velocity can be calculated by knowing, the time it takes for the ball to fall is the same time it takes to travel horizontally from the starting position until the ball hits the floor. The end of the flight is dependent on the vertical motion. Gravitational force, which is vertical, causes the projectile to come to rest at some point. It is this action, which ends the time of flight.Starting with the vertical direction , (1)Knowing the ball was shot horizontally, must equal zero and using the floor as lowest possible point, is also equal to zero. This yields:. (2)Solving for :. (3)Looking at the horizontal direction, (4)where:When the initial velocity can be found using:. (5)Because the ball is fired horizontally, the initial velocity, in the horizontal direction, is equal to the initial velocity Note, this would not be true, if the ball was fired at an angle above or below the horizontal. It should also be noted for problems with small horizontal and vertical dimensions, it is not necessary to consider air resistance. Therefore, the ball, that is shot, will not have a change in the horizontal velocity component until it hits the floor.Ballistic PendulumThe ballistic pendulum uses a ball that is fired from a spring-loaded gun into the bob of the ballistic pendulum, which rises to a final position. The initial velocity, of the ball is determined in terms of the mass of the ball, mass of the pendulum and the height to which the pendulum rises after impact. The momentum of a body is defined as the product of the mass and its velocity : (6)The principle of conservation of momentum follows directly from Newton?s laws of motion. According to this principle, if there are no external forces acting on a system containing several bodies, the momentum of the system remains constant. In this experiment, the principle is applied to the case of a collision using a ballistic pendulum. In a collision between the ball and the pendulum, each one exerts a force on the other. These forces are equal and opposite, and if no other forces are brought into play, the total momentum of the system before the collision is equal to the total momentum after the collision :. (7)The initial momentum comes from the mass of the ball and its initial velocity . The final momentum is found by velocity of the pendulum and ball and their combined masses as shown below:. (8)Solving for the initial velocity of the ball:. (9)In order to solve the equation above, the velocity of the pendulum and ball must be found. Initially, the pendulum is set at rest. The ball strikes the pendulum bob horizontally and remains stuck in it. The impact of the ball puts the pendulum in motion, causing it to swing upward, where it comes to rest some vertical height above the starting position . The collision between the ball and the pendulum is totally inelastic. Hence, immediately after the collision, ball and pendulum move horizontally with the same velocity. An object in motion has energy of motion called kinetic energy . This is the energy an object possesses due to the objects motion and is represented by: . (10)After the collision is over, during the subsequent swinging motion, of the pendulum, the total mechanical energy is conserved. The kinetic energy, of the system, just after the collision, must equal the increase in potential energy, of the system, as the pendulum and ball reach the highest point. The potential energy ) is the amount of stored energy, or latent gravitational work represented by:. (11)This follows from the law of conservation of energy where energy initial equals energy final , assuming the loss of energy due to friction at the point of support is negligible:. (12)At the bottom, of the swing, the energy is all kinetic and at the top, of the swing, it is all potential. Hence, conservation of energy shows: (13)Solving for the velocity :. (14)DirectionsTrajectorySet the ballistic pendulum apparatus near one end of a level table and move the pendulum out of the way, so it will not interfere with the flight of the ball.Get the gun ready for firing, by placing the ball on the end of the firing rod and pushing it back, compressing the spring until the trigger is engaged. Fire the ball toward a cardboard box and observe the approximate location where the ball hits the ground. Then, place a plain white sheet of paper on the floor and secure it in place with tape.Place a piece of carbon paper on top of the white sheet, so when the ball strikes the carbon paper it will leave a mark on the white paper.Fire the apparatus 6-8 times making sure it hits the paper each time.Measure the range of each shot and determine the average distance. This is the horizontal distance the ball traveled.Measure the vertical fall of the ball, that is, the vertical distance of the point of projection above the floor.Ballistic PendulumRelease the pendulum from the rack and allow it to hang freely. When the pendulum is at rest, pull the trigger, thereby firing the ball into the pendulum bob. This will cause the pendulum, with the ball inside it, to swing up along the rack, where it will be caught at its highest point.Measure the vertical distance from the base of the apparatus to the index point attached to the pendulum. This index point indicates the height, of the center of gravity, of the pendulum and ball. Measure to the nearest 0.1 millimeters.Repeat step 2, five to six times and find the average height.Determine the initial height of the index point attached to the pendulum.Calculate the average change in height of the pendulum and ball.Find the mass of the ball and the mass of the pendulum.Calculate the ball and pendulum velocity.Calculate the initial velocity of the ball.Calculate the percent difference between the trajectory data and the ballistic pendulum data.Projectile Motion ResultsTrajectory Data:Change in Height ____________________(m)Mass of the Ball ____________________(kg)Mass of the Pendulum ____________________(kg)Pendulum and Ball Velocity ____________________(m/s)Initial Velocity of the Ball ____________________(m/s)Compare the initial velocity obtained by the two methods: % Difference ____________________