# Momentum Conservation Models - bullet-block, block-board, spring, boat

> Animated classic momentum problems: bullet and block, block on board, spring pair, person and boat, ball on an arc slot, explosions, with energy bars.

- URL: https://qilingtools.com/en/physics/momentum-models
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Six of the most common momentum-conservation models from high-school physics, each with adjustable parameters and an animation: a bullet hitting a block (stopping inside or passing through), a block sliding on a board over a smooth floor, two blocks joined by a spring, a person walking on a boat, a ball riding up a movable smooth arc, and explosions or recoil. Every model shows momentum and energy bars for the start, the key moment, the end and the current instant, plus a v–t graph and a worked solution with numbers.

## How to use

1. Pick a model tab, then a classic preset or set masses, speeds and forces with the sliders
2. Press Play, or jump straight to the key moment (common velocity, highest point)
3. Use the momentum bars to check conservation and the energy bars to see kinetic energy turn into heat, elastic or gravitational energy
4. Read the worked solution, then try the random problems in Practice

## Features

- Six classic models: bullet-block, block-board, spring pair, person-boat, ball and arc slot, explosion/recoil
- Momentum and energy bar charts at four moments
- v–t graph marking common velocity, maximum compression and highest point
- Detects pass-through, sliding off the board and leaving the arc rim automatically
- Worked solutions with numbers, including fixed-block and attached-spring variants
- Random practice problems with instant marking and an animated solution

## FAQ

### Why is the heat in the bullet-block problem Q = f·d?

The resistance does work −f·s₁ on the bullet and +f·s₂ on the block. Together that is −f(s₁ − s₂) = −f·d, where d is the penetration depth (relative displacement). The lost mechanical energy becomes heat, so Q = f·d, which also equals the drop in total kinetic energy.

### When is the elastic energy largest for two blocks joined by a spring?

When both blocks move at the same velocity. Find that common velocity from momentum conservation; the initial kinetic energy minus the kinetic energy at that moment is the maximum elastic potential energy. When the spring returns to natural length the velocities match an elastic collision.

### Where does m₁x₁ = m₂x₂ come from in the person-boat model?

The person and boat feel no horizontal external force and start at rest, so m₁v₁ = m₂v₂ at every instant. Multiplying by time and adding up gives m₁x₁ = m₂x₂. With the geometry x₁ + x₂ = L you get each displacement, whatever the walking speed.

### For a ball on a movable smooth arc, what is conserved?

Mechanical energy of ball plus slot is conserved because there is no friction. Only horizontal momentum is conserved: the floor pushes up, so vertical momentum is not. At the highest point both share the same horizontal speed, and when the ball leaves the bottom the result is the same as an elastic collision.

### Why can momentum conservation be used for an explosion?

An explosion is very short and its internal forces are far larger than gravity or drag, so the external impulse is negligible. Momentum is therefore conserved across it, while chemical energy turns into extra kinetic energy, unlike in a collision.

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