Inelastic Collision Formula

Inelastic collision is a collision in which the total momentum of the system is conserved but the total kinetic energy is not — some kinetic energy is.

The Formula

m1v1+m2v2=(m1+m2)vfm_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f (perfectly inelastic)

When to use: Two cars crashing and sticking together: they move as one object and kinetic energy is lost.

Quick Example

A ball of clay hitting a wall and sticking — it doesn't bounce; the kinetic energy converts to deformation.

Notation

m1,m2m_1, m_2 are the masses in kg, v1,v2v_1, v_2 are the initial velocities in m/s, vfv_f is the common final velocity in m/s, and ΔKE\Delta KE is the kinetic energy lost in joules.

What This Formula Means

A collision in which the total momentum of the system is conserved but the total kinetic energy is not — some kinetic energy is converted.

Two cars crashing and sticking together: they move as one object and kinetic energy is lost.

Formal View

For a perfectly inelastic collision: m1v⃗1+m2v⃗2=(m1+m2)v⃗fm_1 \vec{v}_1 + m_2 \vec{v}_2 = (m_1 + m_2)\vec{v}_f. The kinetic energy lost is ΔKE=12m1m2m1+m2(v1−v2)2\Delta KE = \frac{1}{2}\frac{m_1 m_2}{m_1 + m_2}(v_1 - v_2)^2, which is always positive.

Worked Examples

Example 1

medium
A 0.04 kg0.04 \text{ kg} bullet at 300 m/s300 \text{ m/s} embeds in a 1.96 kg1.96 \text{ kg} block at rest. Find vfv_f.

Answer

vf=6 m/sv_f = 6 \text{ m/s}

First step

1
0.04⋅300=(0.04+1.96)vf=2vf0.04 \cdot 300 = (0.04 + 1.96) v_f = 2 v_f.

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Example 2

medium
A 1 kg1 \text{ kg} ball at 10 m/s10 \text{ m/s} hits a 3 kg3 \text{ kg} ball at rest; they stick. Find vfv_f and the fraction of KE retained.

Example 3

hard
Ballistic pendulum: a 0.01 kg0.01 \text{ kg} bullet embeds in a 0.99 kg0.99 \text{ kg} pendulum at rest. The combined block rises 0.45 m0.45 \text{ m} (g=10 m/s2g = 10 \text{ m/s}^2). Find the bullet's initial speed.

Common Mistakes

  • Trying to use conservation of kinetic energy — in inelastic collisions, kinetic energy is NOT conserved; only momentum is. - 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 to include both objects' momenta before the collision — if one object is initially at rest, its momentum is zero but it still has mass that affects the final velocity. - 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.
  • Confusing 'inelastic' with 'perfectly inelastic' — in a perfectly inelastic collision the objects stick together (maximum KE loss), but ordinary inelastic collisions lose some KE without sticking. - 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 inelastic 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

Inelastic 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 Inelastic Collision formula?

A collision in which the total momentum of the system is conserved but the total kinetic energy is not — some kinetic energy is converted.

How do you use the Inelastic Collision formula?

Two cars crashing and sticking together: they move as one object and kinetic energy is lost.

What do the symbols mean in the Inelastic Collision formula?

m1,m2m_1, m_2 are the masses in kg, v1,v2v_1, v_2 are the initial velocities in m/s, vfv_f is the common final velocity in m/s, and ΔKE\Delta KE is the kinetic energy lost in joules.

Why is the Inelastic Collision formula important in Physics?

Inelastic 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 Inelastic Collision?

Students often know a formula related to inelastic 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 Inelastic Collision formula?

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

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