力学 Classical Mechanics · Laboratory experiment

The Ballistic Pendulum: Momentum & Energy Conservation

Aim: to determine the muzzle velocity \(v_0\) of a high-speed projectile by inelastic collision with a suspended pendulum catcher, and to cross-verify the velocity independently using free projectile flight.

Apparatus & Trajectory

Cocked & Ready
Click "Fire Gun" to launch the projectile. The ratchet needle tracks the peak angle \(\theta_{\max}\).

Energy Partition & Inelastic Loss

Kinetic energy before: \(\frac12 m v_0^2\). Only \(\frac{m}{m+M}\) survives the impact into pendulum kinetic energy; the remaining \(\frac{M}{m+M}\) turns into acoustic deformation heat.

Independent Cross-Check: Free Projectile

Swing the pendulum bob up out of the trajectory, allowing the projectile to fly freely off the laboratory bench of height \(H\). Measuring range \(X\) provides an independent test:

\(v_0 = X \sqrt{\frac{g}{2H}}\)

Observation Table

Theory: Inelastic Collision & Angular Swing

The ballistic pendulum consists of a spring gun and a suspended mass catcher of mass \(M\) hanging at radius \(R\) from a low-friction bearing. When a projectile of mass \(m\) is fired horizontally with velocity \(v_0\) into the catcher, two distinct physical stages unfold:

  1. Stage 1: Inelastic Impact (Time scale \(\Delta t \approx 10^{-4}\) s).
    Because the collision is instantaneous, external forces (gravity and pivot tension) impart negligible impulse during the contact. Linear momentum is conserved: $$ m v_0 = (m + M) V_1 \implies V_1 = \frac{m}{m + M} v_0 $$ Mechanical energy is not conserved during impact; a fraction \(\frac{M}{m+M}\) converts into acoustic vibrations and heat.
  2. Stage 2: Conservative Swing.
    After the ball is captured, the combined mass swings upward until its kinetic energy converts completely into gravitational potential energy at apex angle \(\theta_{\max}\): $$ \frac{1}{2}(m + M) V_1^2 = (m + M) g \Delta h = (m + M) g R (1 - \cos\theta_{\max}) $$ $$ V_1 = \sqrt{2 g R (1 - \cos\theta_{\max})} = 2 \sqrt{g R} \sin\left(\frac{\theta_{\max}}{2}\right) $$

Equating expressions for \(V_1\) yields the initial muzzle velocity:

$$ v_0 = \left(\frac{m + M}{m}\right) \sqrt{2 g R (1 - \cos\theta_{\max})} $$

Experimental Procedure & Viva Questions

  1. Measure the projectile mass \(m\) and bob mass \(M\) with the laboratory balance.
  2. Set the spring gun to Notch 2. Release the trigger and observe the drag pointer push across the circular protractor to record \(\theta_{\max}\).
  3. Perform 5 trials to quantify mechanical scatter and uncertainty.
  4. Swing the bob up to latch it clear of the line of fire. Fire 5 free projectile shots to the floor pad to measure range \(X\). Compare the ballistic \(v_0\) with kinematic \(v_0\).

Viva Questions

Why can we not apply conservation of kinetic energy during the collision?

The ball deforms the catcher cavity, generating internal friction, heat, and sound. Mechanical energy is dissipative during the crash.

What is the purpose of the ratchet / friction needle?

The pendulum swings back down immediately after reaching apex. The lightweight friction pointer is pushed forward and stays at the maximum deflection angle, eliminating visual tracking error.

How does the center of mass shift when the ball embeds?

The effective radius \(R\) is measured from the pivot axis to the combined center of mass of the catcher plus embedded projectile.