A ballistic pendulum is a classic device for finding the speed of a fast projectile, such as a bullet, by letting it embed in a hanging block. The collision is brief and highly inelastic, so the projectile and block move together immediately afterward. This makes the system easier to analyze because the fast motion is converted into a slower swing that can be measured.
The idea matters because it shows how momentum and energy conservation can be used in different parts of the same physical event.
The analysis has two stages: the collision and the upward swing. During the collision, momentum is conserved because the external impulse from gravity and the string tension is very small over the short collision time, but kinetic energy is not conserved. During the swing after impact, mechanical energy is conserved if air resistance and pivot friction are negligible, so the combined mass trades kinetic energy for gravitational potential energy.
By combining these two conservation laws, the original projectile speed can be calculated from the rise height of the block-projectile system.
Key Facts
- Inelastic collision stage: m v = (M + m) V
- Swing energy stage: 1/2 (M + m) V^2 = (M + m) g h
- Speed just after impact: V = sqrt(2 g h)
- Projectile speed: v = ((M + m) / m) sqrt(2 g h)
- Height from swing angle: h = L(1 - cos theta)
- The collision conserves momentum, while the swing conserves mechanical energy.
Vocabulary
- Ballistic pendulum
- A device that measures projectile speed by capturing the projectile in a hanging block and observing the height of the swing.
- Perfectly inelastic collision
- A collision in which objects stick together and move with a common velocity after impact.
- Momentum
- The quantity of motion of an object, equal to mass times velocity, p = mv.
- Mechanical energy
- The total kinetic energy and gravitational potential energy of a system.
- Rise height
- The vertical distance the block-projectile system rises after the collision.
Common Mistakes to Avoid
- Using energy conservation during the collision is wrong because the projectile embeds in the block and mechanical energy is lost to heat, sound, and deformation.
- Forgetting to include both masses after impact is wrong because the block and projectile move together as one combined system with mass M + m.
- Using the final swing height as the projectile height is wrong because h is the vertical rise of the combined block-projectile center of mass after impact.
- Confusing V with v is wrong because V is the slower speed just after impact, while v is the much larger projectile speed before impact.
Practice Questions
- 1 A 0.020 kg projectile embeds in a 1.50 kg block. After impact, the combined system rises 0.080 m. Find the projectile's initial speed using g = 9.8 m/s^2.
- 2 A 0.010 kg pellet strikes a 0.490 kg pendulum block and sticks. The pendulum length is 0.75 m and the maximum swing angle is 18 degrees. Find the pellet's initial speed using h = L(1 - cos theta).
- 3 Explain why momentum conservation is applied to the collision stage but mechanical energy conservation is applied only to the swing stage.