Elastic Collision Formula

Elastic collision is a collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact.

The Formula

pi=pf and KEi=KEfp_i = p_f \text{ and } KE_i = KE_f

When to use: Billiard balls bouncing off each other: the total energy stays the same, nothing is lost to heat or deformation.

Quick Example

A steel ball bearing bouncing off another of equal mass — the first stops, the second moves at the same speed.

Notation

m1,m2m_1, m_2 are the masses, v1i,v2iv_{1i}, v_{2i} are initial velocities, v1f,v2fv_{1f}, v_{2f} are final velocities, pp is momentum in kg·m/s, and KEKE is kinetic energy in joules.

What This Formula Means

A collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact.

Billiard balls bouncing off each other: the total energy stays the same, nothing is lost to heat or deformation.

Formal View

In an elastic collision: ∑mivi,before=∑mivi,after\sum m_i v_{i,\text{before}} = \sum m_i v_{i,\text{after}} and ∑12mivi,before2=∑12mivi,after2\sum \frac{1}{2}m_i v_{i,\text{before}}^2 = \sum \frac{1}{2}m_i v_{i,\text{after}}^2. Equivalently, the relative velocity reverses: v1i−v2i=−(v1f−v2f)v_{1i} - v_{2i} = -(v_{1f} - v_{2f}).

Worked Examples

Example 1

medium
A 2 kg2\,\text{kg} ball at 5 m/s5\,\text{m/s} hits a 4 kg4\,\text{kg} ball at rest elastically. Find v1′v_1' using v1′=m1−m2m1+m2v1v_1' = \frac{m_1 - m_2}{m_1 + m_2}v_1.

Answer

v1′=−53 m/s≈−1.67 m/sv_1' = -\tfrac{5}{3} \text{ m/s} \approx -1.67 \text{ m/s}

First step

1
Plug in: v1′=2−42+4(5)=−26(5)v_1' = \frac{2 - 4}{2 + 4}(5) = \frac{-2}{6}(5).

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Example 2

medium
A 0.2 kg0.2\,\text{kg} ball at 10 m/s10\,\text{m/s} hits a 0.6 kg0.6\,\text{kg} ball at rest elastically. Find both final velocities.

Example 3

hard
Show: in an elastic 1D collision with m2≫m1m_2 \gg m_1 and v2=0v_2 = 0, the light particle bounces back at speed ∣v1∣|v_1| unchanged.

Common Mistakes

  • Assuming all bouncing collisions are perfectly elastic — most real collisions lose some kinetic energy to heat, sound, or deformation, even if objects bounce apart. - Fix this by naming the system, checking "Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?", and attaching units or direction to the final statement.
  • Using only conservation of momentum and neglecting the kinetic energy equation — elastic collisions require both conservation laws to solve for two unknowns. - Fix this by naming the system, checking "Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?", and attaching units or direction to the final statement.
  • Forgetting the shortcut: in elastic collisions, v1i−v2i=−(v1f−v2f)v_{1i} - v_{2i} = -(v_{1f} - v_{2f}), which can replace the energy equation and simplify algebra. - Fix this by naming the system, checking "Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?", and attaching units or direction to the final statement.
  • Using elastic collision from a keyword alone - Signal words like momentum, impulse, collision only point to a possible model; the system must match too.

Why This Formula Matters

Elastic Collision is central because forces explain changes in motion and balance. Students who can isolate a system and draw the interactions can avoid treating every force word as the same kind of cause.

Frequently Asked Questions

What is the Elastic Collision formula?

A collision in which both the total momentum and the total kinetic energy of the system are fully conserved after impact.

How do you use the Elastic Collision formula?

Billiard balls bouncing off each other: the total energy stays the same, nothing is lost to heat or deformation.

What do the symbols mean in the Elastic Collision formula?

m1,m2m_1, m_2 are the masses, v1i,v2iv_{1i}, v_{2i} are initial velocities, v1f,v2fv_{1f}, v_{2f} are final velocities, pp is momentum in kg·m/s, and KEKE is kinetic energy in joules.

Why is the Elastic Collision formula important in Physics?

Elastic Collision is central because forces explain changes in motion and balance. Students who can isolate a system and draw the interactions can avoid treating every force word as the same kind of cause.

What do students get wrong about Elastic Collision?

Students often know a formula related to elastic collision but skip the recognition step: Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored? That leads to a correct-looking substitution attached to the wrong physical model.

What should I learn before the Elastic Collision formula?

Before studying the Elastic Collision formula, you should understand: conservation of momentum, kinetic energy.

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