How Momentum Is Conserved in Collisions

Quick Answer

In every collision the total momentum is conserved: the sum p = m1 v1 + m2 v2 is the same before and after the impact. Kinetic energy, though, is only conserved in an elastic collision. For example, two equal carts in an elastic collision swap velocities, while in a perfectly inelastic collision they stick together and lose some kinetic energy to heat. The masses and the coefficient of restitution set the outcome. Open this simulator, pick the masses and speed, and watch momentum hold steady while the carts collide.

How a Collision Works

The canvas shows two carts on a frictionless track. Cart 1 rolls in and strikes cart 2, and their velocities change in an instant according to conservation of momentum and the restitution you set. For example, that is why momentum is conserved: the push cart 1 gives cart 2 is exactly equal and opposite to the push it feels back, so the total momentum p = m1 v1 + m2 v2 cannot change. The velocity graph shows both carts trading speed at the moment of impact.

Momentum vs Kinetic Energy

Two quantities describe the collision, and they behave differently.

• Momentum (p = m v): always conserved. The total before the collision equals the total after, whatever the restitution.
• Kinetic energy (half m v squared): conserved only in an elastic collision. An inelastic collision loses some to heat and sound.
• Restitution (e): sets how bouncy the impact is, from e = 1 (perfectly elastic) down to e = 0 (perfectly inelastic, carts stick).

Calculating the Outcome

To calculate the momentum after a collision, add up mass times velocity for both carts; it equals the value from before, so nothing new is needed. To find the final velocity of each cart, combine conservation of momentum with the restitution relation and solve for the two unknowns. For example, a 5 kg cart at 4 m/s striking a 1 kg cart elastically sends the light cart off near 6.7 m/s while the heavy cart slows only slightly, and the simulator reads both velocities straight off the carts.

Elastic vs Inelastic Collisions

An elastic collision, e = 1, conserves kinetic energy as well as momentum, so the carts rebound cleanly like a Newton's cradle. An inelastic collision loses kinetic energy: at e = 0 the carts stick together and move as one, as in a car crash test where the metal crumples. For example, comparing elastic vs inelastic in the simulator shows the momentum reading unchanged in both cases while the kinetic energy reading drops only when e is below 1.

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