COLLEGE PHYSICS LABORATORY EXPERIMENT

BALLISTIC PENDULUM

THEORY

The muzzle velocity v0 of a projectile launcher is determined by launching a ball of massm into a pendulum of mass M and observing the angle θ to which the pendulum swings.

The process can be divided into two parts. Part I involves the collision of the ballwith the pendulum. Since the ball and the pendulum become combined after the collision,this is an inelastic collision and, therefore, only momentum conservation applies. Part IIinvolves the conversion of kinetic energy into gravitational potential energy in the absenceof friction and applied forces.

• Part I: Conservation of MomentumThe momentum before the collision of the ball of mass m and velocity v0 with the pendulumof mass M (at rest) is pbefore = m v0 while the momentum of the combined ball-pendulumsystem (travelling at velocity V ) after the collision is pafter = (m + M) V . Momentumconservation implies that pbefore = pafter or

m v0 = (m + M) V ⇒ v0 =(1 +

M

m

)V. (1)

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Hence, the muzzle velocity v0 of the projectile launcher can be determined from V and themass ratio M/m.

• Part II: Energy ConversionImmediately after the collision, the combined ball-pendulum system travels horizontallyat a velocity V . Its initial kinetic energy is therefore Kinitial = 1

2(m + M) V 2. Assuming

that at the moment of collision, the pendulum was at its lowest vertical position (wherewe may set the gravitational potential energy equal to zero), the initial potential energyof the ball-pendulum system is, therefore, Uinitial = 0. In the process of energy conversion,the initial kinetic energy is converted into potential energy until the final kinetic energyis zero. According to the law of conservation of mechanical energy (E = K + U), thefinal gravitational potential energy reaches a maximum value Ufinal = Einitial. Using theexpression for the near-Earth gravitational potential energy U = mgh (where g = 9.8 m/s2

denotes the gravitational acceleration), the final potential energy can be expressed in termsof a maximum height h above the lowest vertical position. As the pendulum swings overan angle θ (measured from the vertical), this maximum height is expressed as

h = R (1 − cos θ) ,

where R is the distance from the pivot point to the center of mass of the ball-pendulumsystem. Lastly, using the law conservation of energy, we find

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2(m + M) V 2 = (m + M) gR (1 − cos θ) ⇒ V =

√2gR (1 − cos θ). (2)

By combining Eqs. (1) and (2), we find the muzzle velocity

v0 =(1 +

M

m

) √2gR (1 − cos θ)

MATERIAL & PROCEDURE

• Material: Ballistic Pendulum & Projectile Launcher (with instructions) and massbalance.

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• Procedure: ◦ Follow Setup instructions.◦ Choose ball mass m and pendulum mass M so that the final

angle θ is lower than maximum angle allowed (θ < 90o).◦ Remove pendulum from the base and find masses of the ball and

the combined ball-pendulum system.◦ Measure the distance from the pivot point to the balance point

of the combined ball-pendulum system.◦ Replace the pendulum in the base and load the launcher with the ball.◦ Set the angle indicator to an angle 1 − 2o less than that

reached in the calibration step.◦ Fire the launcher and record the angle reached by the ball-pendulum

system. Repeat several times for each pendulum mass M .◦ Repeat steps for different values of pendulum mass M .

• Design an experiment involving projectile motion to independentlymeasure the muzzle velocity.

DATA ANALYSIS

• Find average angle θ (and its mean deviation δθ) reached by the ball-pendulumsystem for each pair (m, M).

• Calculate the average muzzle velocity v0 and its mean deviation for the ballisticpendulum experiment.

• Calculate the average muzzle velocity v0 and its mean deviation for the projectilemotion experiment.

• Compare your two experimental values for the muzzle velocity by computing theirpercent difference.

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