Ballistic pendulum as illustrated in Figure 1 below is a classical method of measuring the speed of bullets. This technique, which is commonly used in investigation of bullet speeds, relies heavily on the basic laws of conservation in physics.
Figure 1 Ballistic Pendulum Set-up
Figure 2 illustrates how the experiment is executed chronologically from the time the ‘bullet’ is launched to the time it ceases travelling at the maximum angular displacement. In the illustration, the bullet of mass, m and initial velocity, v hits a stationary pendulum cup and it is embedded in it to form a mass, M. The speed at which collision takes place is sufficiently fast such that both m and M are viewed as a closed system. After collision, the formed composite of mass, m+M has a new velocity of, v0. The composite mass acts as a pendulum and it moves upwards, using its initial kinetic energy to overcome the gravity force and thereby being converted to potential energy. The total amount of potential energy gained in the process of swinging is equal to the kinetic energy lost at the point of halt. The gained potential energy is given by:
Whereby h is the height, over which the mass center rises. In this experiment, the muzzle velocity of a bullet is obtained by measuring the vertical displacement, h, which is indicative of the gained potential energy following the collision. From this information, it ia possible to predict the range of the steel ball had the it not been captured by the ballistic pendulum.
- rb, Distance from the pivot point to the center of the ball just before it enter pendulum cup.
- m-Mass of the ball
- M-Mass of the pendulum and the ball
- rCM-Distance from the pivot to the pendulum center of mass
- Δt-Time for the ball to move through the gate
- Angle of rotation in degrees
These data are summarized in the table below:
Thus, the initial velocity is given by:
Initial velocity, v0=mp+mbvrmb
Kinetic Energy before collision=Potential Energy after collision
Thus, recoil velocity is given by:
The velocity of the bullet may be obtained from the formula below: