Click here to see a video of the first part of the experiment

An interesting problem that arises in physics is how to measure the speed of a small, fast projectile such as a bullet. Obviously, it goes too fast to use a stopwatch accurately, and it is too small to use a radar system. What is needed is some convenient, reasonably accurate, perhaps even portable device for making such measurements. One such device is the ballistic pendulum. While the apparatus used in this laboratory exercise cannot be used for actual bullets, the principles which govern its behavior are exactly the same as those for any ballistic pendulum.

The principle of conservation of momentum follows directly from Newton’s law of motion. According to this fundamental principle, if there are no external forces acting on a system containing several bodies, then the momentum of the system remains constant. In this experiment the principle is applied to the case of a collision, using a ballistic pendulum. A ball is fired by a gun into the pendulum’s bob. The initial velocity of the ball is determined in terms of the masses of the ball and the bob and the height to which the bob rises after impact. This velocity can also be obtained by firing the ball horizontally and allowing it to fall freely toward the earth. The velocity is then determined in terms of the range and the vertical distance of fall.

When you have completed this experimental activity you should be able to: (1) determine the time of flight of a horizontally projected object by measuring its distance of fall; (2) determine the velocity of a projectile by measuring its range and fall; (3) distinguish between elastic and inelastic collisions; (4) apply the principle of conservation of momentum to an inelastic collision; and (5) determine the velocity of a projectile using a ballistic pendulum.

The momentum of a body is defined as the product of the mass of the body and its velocity. Newton’s second law of motion states that the net force acting on a body is proportional to the time rate of change of momentum. Hence, if the sum of the external forces acting on a body is zero, the linear momentum of the body is constant. This is essentially a statement of the principle of conservation of momentum. Applied to a system of bodies, the principle states that if no external forces act on a system containing two or more bodies, then the momentum of the system does not change.

In a collision between two bodies, 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 two bodies is not changed by the impact. Hence the total momentum of the system after collision is equal to the total momentum of the system before collision. During the collision the bodies become deformed and a certain amount of energy is used to change their shape. If the bodies are perfectly elastic, they will recover completely from the distortion and will return all of the energy that was expended in distorting them. In this case, the total kinetic energy of the system remains constant. If the bodies are not perfectly elastic, they will remain permanently distorted, and the energy used up in producing the distortion is not recovered.

Inelastic impact can be illustrated by a device known as the ballistic pendulum, which is sometimes employed in determining the speed of a bullet. If a bullet is fired into a pendulum bob and remains embedded in it, the momentum of the bob and bullet just after the collision is equal to the momentum of the bullet just before the collision. This follows from the law of conservation of momentum. The velocity of the pendulum before collision is zero, while after the collision the pendulum and the bullet move with the same velocity. Hence the momentum equation gives:

where m is the mass of the bullet, v is the velocity of the bullet just before the collision, M is the mass of the pendulum bob, and V is the common velocity of the bob and bullet just after the collision.

As a result of the collision, the pendulum with the embedded bullet swings about its point of support, and the center of gravity rises through a vertical distance h. From the measurement of this distance it is possible to calculate the velocity V in the following way: As a pendulum swings, its total mechanical energy, i.e., kinetic energy plus potential energy, must remain constant (neglecting any friction losses). Therefore, the kinetic energy of the pendulum at its lowest point of swing must equal the potential energy at the highest point of swing. This results in the following energy equation:

where h is the vertical rise of the center of gravity of the bob with bullet, and g is the acceleration of gravity. The left side of equation (2) is the kinetic energy term, and the right side is the potential energy expression. If you solve this equation for V, you get:

The velocity of the bullet can also be determined from measurements of the range and vertical distance of fall when the bullet is fired horizontally and allowed to fall to the floor without striking the pendulum bob.

The motion of a projectile is a special case of a freely falling body in which the initial velocity may be in any direction with respect to the vertical. The path of the projectile is a curved path produced by a combination of the uniform velocity of projection and the velocity due to the acceleration of gravity. This type of motion may be studied very effectively by considering it as made up of two independent motions, one of constant speed in the horizontal direction, and the other of constant acceleration in the vertical direction.

For the special case where the projectile is fired horizontally with an initial velocity v, the equations are extremely simple, as will be seen in equations (4) and (5). Since the horizontal component of the velocity remains constant, the distance traveled in the horizontal direction is given by:

where v is velocity and t is time. Because the bullet is projected horizontally, the initial velocity in the vertical direction is zero, and the resulting equation for the vertical position is:

This equation can be rearranged to determine the time taken to fall through a known vertical distance y, as:

In the first part of the experiment, the initial velocity of the projectile will be determined by measurements of range and fall. The apparatus should be set near one edge of a level table. In this part the pendulum is not used, and can be removed from its pivots so that it will not interfere with the free flight of the ball. Prepare the gun 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. BE CAREFUL with the gun, as the projectile could cause pain or injury, and the gun mechanism is unforgiving to fingers that get too close during firing.

The ball should be fired horizontally so that it strikes a target placed on the floor. Be sure that the ball does not hit the table, or wall, or anything else before striking the floor. Make a test firing to determine approximately where the ball will hit the floor, then place some of the automatic carbon paper at that location. The paper should be taped down so that it will not move. The carbon paper will give a permanent record of the location of "hits’ when the gun is fired. Make a series of at least ten shots, all of which should fall relatively close to each other. Make sure that the apparatus does not move between firings. You may want to mark its position. Also, be careful to always compress the spring to the same position. The ballistic pendulum may have more than one "cocked" position for the spring.

Determine the point on the floor directly beneath the release point of the gun. Find this point as carefully as you can, even though some estimation is required. Put some masking tape at this point, and mark on the masking tape rather than the floor. Now measure from this point to each of the impact points, and record these distances, which are the ranges of each shot. Also, measure and record the vertical distance of fall of the ball from its point of release to the floor.

Using the distance of vertical fall of the projectile, compute the time of fall with equation (6). Check to see that the value you find for the time is a reasonable value, and not excessively large or small. Then find the average of the ten range measurements, and determine the initial velocity from equation (4), using the time you just determined and the average range value.

Find the average notch of your ten ballistic pendulum firings, and place the pendulum in that notch. Measure the height of the center of mass above the base of the apparatus when the pendulum is on the rack, and then measure the height of the center of mass when the pendulum hangs straight downward. Use these two measurements to determine the rise of the pendulum during firing. The center of mass of the pendulum is determined by placing the ball in the pendulum and balancing the entire pendulum on a narrow edge, such as the edge of a ruler. The position at which it balances is the center of mass. You may place a piece of tape on the pendulum arm and mark the center of mass, but remove the tape when you are finished.

From equation (3), the velocity of the loaded pendulum can be determined, and this value can be substituted into equation (1) to find the value of the bullet’s initial velocity. This latter value should be close to the value of velocity found from the range and fall measurements, and since they are independent determinations of the same value, you can compare them with a percent difference.

As a final item, compare the kinetic energy of the ball before collision with the kinetic energy of the ball and bob after collision, to see if the collision is elastic or inelastic. What percent of the initial kinetic energy of the ball remains after the collision? What conclusions can you draw to account for any losses?

Report you average value of velocity of the projectile and discuss the merits of the ballistic pendulum as an experimental apparatus. Also, report your values for the kinetic energy before the collision and the kinetic energy after the collision. Is the collision elastic or inelastic? How do you know? What percent of the kinetic energy remains after the collision? What conclusions can you draw to account for the loss?

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