Unit 4.3 | Collisions and Explosions

Conceptual Practice

Three quick checks on what is conserved, before any math.

Problem Type 1: Stick together

How to spot it: “Stick”, “lock together”, “embed”, “move off as one”.

Setup: [katex] m_1v_1 + m_2v_2 = (m_1 + m_2)v’ [/katex]. One unknown, one equation. If the question asks about energy, compare kinetic energy before and after.

A [katex] 1200\,\text{kg} [/katex] car moving at [katex] 15.6\,\text{m/s} [/katex] hits a stationary [katex] 1500\,\text{kg} [/katex] car. They lock together. How fast do they move, and how much kinetic energy is lost?

Momentum: [katex] (1200)(15.6) + 0 = (2700)v’ \rightarrow v’ \approx 6.93 \, m/s [/katex]

Energy before: [katex] \frac{1}{2}(1200)(15.6)^2 \approx 146{,}000 \, J [/katex]. After: [katex] \frac{1}{2}(2700)(6.93)^2 \approx 65{,}000 \, J [/katex]. About [katex] 81{,}000\,\text{J} [/katex] (more than half) went into crumpling metal, heat and sound.

AP trap: Using conservation of kinetic energy to find the final speed. Objects that stick always lose kinetic energy, so only momentum works.

Problem Type 2: Bounce apart

How to spot it: “Bounces”, “rebounds”, “elastic”, or two different final velocities.

Setup: Momentum gives one equation with signs. To decide if the collision was elastic, compare the total kinetic energy before and after.

A [katex] 2\,\text{kg} [/katex] cart at [katex] 6\,\text{m/s} [/katex] hits a [katex] 4\,\text{kg} [/katex] cart at rest. Afterward the [katex] 2\,\text{kg} [/katex] cart rolls back at [katex] 2\,\text{m/s} [/katex]. How fast does the [katex] 4\,\text{kg} [/katex] cart move? Was the collision elastic?

Momentum (right is positive): [katex] (2)(6) + 0 = (2)(-2) + (4)v \rightarrow v = 4 \, m/s [/katex].

Energy before: [katex] \frac{1}{2}(2)(6)^2 = 36 \, J [/katex]. After: [katex] \frac{1}{2}(2)(2)^2 + \frac{1}{2}(4)(4)^2 = 4 + 32 = 36 \, J [/katex]. No kinetic energy lost, so it was elastic.

AP trap: Dropping the minus sign on the cart that bounces back. With it, you get the right answer; without it, the [katex] 4\,\text{kg} [/katex] cart would have to move at [katex] 2\,\text{m/s} [/katex] and the energy check fails.

Problem Type 3: Explosions and recoil

How to spot it: Everything starts at rest (or moving together) and flies apart: guns, skaters pushing off, a spring between two carts, a firework.

Setup: The total momentum before is zero (or one shared value), so [katex] 0 = m_1v_1′ + m_2v_2′ [/katex]. The pieces leave with equal and opposite momentum.

A [katex] 45\,\text{kg} [/katex] skater and a [katex] 60\,\text{kg} [/katex] skater push off each other from rest. The [katex] 45\,\text{kg} [/katex] skater moves at [katex] 0.375\,\text{m/s} [/katex]. How fast does the [katex] 60\,\text{kg} [/katex] skater move?

[katex] 0 = (45)(-0.375) + (60)v \rightarrow v = 0.28 \, m/s [/katex] in the opposite direction. The heavier skater moves slower, but with the same size momentum.

Guns recoil, rockets fly and swimmers push off walls for the same reason.

AP trap: Thinking an explosion breaks conservation of momentum because kinetic energy appears from nowhere. The kinetic energy comes from chemical or spring energy; the momentum still adds to zero.

Lesson 4.4 Preview

In the final lesson of this unit we chain momentum and energy together for the hardest problems: a collision followed by a swing, a slide or a spring. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-4-3-collisions-and-explosions/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-4-3-collisions-and-explosions/