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

How to use Momentum Conservation Models

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

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.