Ballistic Pendulum & Spring Systems: Calculating Amplitude and Velocity

To solve a ballistic spring problem where a bullet embeds in a block, you must separate the event into two distinct phases. First, apply Conservation of Momentum to find the combined velocity immediately after the perfectly inelastic collision (v_f). Second, use Conservation of Energy to convert the system's kinetic energy into elastic potential energy to find the amplitude (A).

The "Novel Insight" Unlike standard textbook definitions that blur the lines, this lecture utilizes the "Two Distinct Phases" framework. This clarifies that the collision occurs so instantly that spring compression is negligible, preventing the common error of applying spring forces during the momentum calculation phase.

"Because the problem tells us the bullet embeds itself into the block, we know right away this is a 'perfectly inelastic' collision." "At the point of maximum compression—which is the amplitude A—the block stops for a tiny instant."

Key Concepts

• Perfectly Inelastic Collision: The bullet and block merge, conserving momentum but losing kinetic energy.
• Momentum (p): Calculated as m · v = (m + M) · v_f.
• Elastic Potential Energy (PE_spring): Stored energy at max compression, calculated as ½ k A².
• Combined Mass: The crucial mass value (m + M) used for the post-collision energy phase.

Frequently asked questions (FAQ)

Q: Why can't I set the bullet's initial KE equal to the spring's PE? A: Energy is not conserved during the collision. A significant amount of the bullet's initial kinetic energy is lost to heat and deformation when it embeds into the block. You must bridge the gap using momentum first.

Q: Do I include the bullet's mass in the spring compression phase? A: Yes. Once embedded, the new system mass is the sum of the block and the bullet (m + M). Failing to add them is a common calculation error.

Q: Why is "negligible compression" important? A: It implies the spring hasn't started pushing back yet during the split-second of impact. This allows us to assume no external forces are acting horizontally, making the Conservation of Momentum equation valid.