MotionLab

Elastic Collision Calculator

Both momentum and kinetic energy are conserved — solve for both final velocities in one step.

Mass 1 (kg)

Velocity 1 (m/s)

Mass 2 (kg)

Velocity 2 (m/s)

v₁′ (m/s)

-3.4

Conservation of momentum + KE

v₂′ (m/s): 3.6
KE total (J): 31

What you entered

Mass 1 (kg): 2Velocity 1 (m/s): 5Mass 2 (kg): 3Velocity 2 (m/s): -2

1
v₁′ = [(m₁−m₂)/(m₁+m₂)]v₁ + [2m₂/(m₁+m₂)]v₂
[(2−3)/(2+3)]×5 + [2×3/(2+3)]×-2= -3.4 m/s
2
v₂′ = [2m₁/(m₁+m₂)]v₁ + [(m₂−m₁)/(m₁+m₂)]v₂
[2×2/(2+3)]×5 + [(3−2)/(2+3)]×-2= 3.6 m/s
3
KE initial = ½m₁v₁² + ½m₂v₂²
0.5×2×5² + 0.5×3×-2²= 31 J
4
KE final (conserved in elastic collision)
0.5×2×-3.4² + 0.5×3×3.6²= 31 J

Result

v₁′ (m/s): -3.4

After the elastic collision: object 1 (2 kg) moves at -3.4 m/s, object 2 (3 kg) at 3.6 m/s. Total KE conserved at 31 J.

Elastic collisions: velocities before and after

In a perfectly elastic collision both momentum and kinetic energy are conserved. Negative velocities mean motion in the opposite direction.

m1 (kg)	v1 (m/s)	m2 (kg)	v2 (m/s)	v1 after	v2 after	KE before (J)	KE after (J)
1	5	1	0	0.000	5.000	12.500	12.500
2	5	1	0	1.667	6.667	25.000	25.000
1	5	2	0	-1.667	3.333	12.500	12.500
1	5	1	-5	-5.000	5.000	25.000	25.000
5	2	3	-1	-0.250	2.750	11.500	11.500

The first row is the Newton's cradle result: equal masses exchange velocities exactly, so the moving ball stops dead and the stationary one departs at the original speed. The third row shows a light object striking a heavier one and bouncing back with a negative velocity, while in the second a heavy object hitting a light one continues forward. The last two columns are identical in every row, which is the definition of elastic - real collisions lose energy to heat, sound and deformation, so this is an idealisation that fits billiard balls and gas molecules far better than cars.

What makes a collision 'elastic'

In a perfectly elastic collision, total kinetic energy is conserved — no energy is lost to heat, sound, or deformation. Billiard balls are approximately elastic; car crashes are highly inelastic. The distinction determines whether you need one conservation law (momentum only) or two (momentum + energy).

Newton's cradle: the classic demonstration

Newton's cradle (the desk toy with swinging steel balls) demonstrates elastic collisions beautifully — when one ball strikes, it stops and the ball on the opposite end flies out at the same speed, simultaneously conserving both momentum and kinetic energy.

Frequently asked questions

A 2 kg ball moving at 5 m/s hits a stationary 3 kg ball head-on. What are the final velocities?

v₁' = (m₁−m₂)/(m₁+m₂) × v₁ = (2−3)/5 × 5 = −1 m/s (bounces back). v₂' = 2m₁/(m₁+m₂) × v₁ = 4/5 × 5 = 4 m/s. Check: momentum before = 10, after = −2 + 12 = 10. KE before = 25 J, after = 1 + 24 = 25 J.

What happens when two equal-mass objects collide elastically?

They exchange velocities. A 1 kg ball at 5 m/s hitting a stationary 1 kg ball stops completely while the second ball takes off at 5 m/s. This is exactly what Newton's cradle demonstrates — one ball in, one ball out, at the same speed.

How is an elastic collision different from an inelastic one?

In an elastic collision, both momentum and kinetic energy are conserved — no energy is lost. In an inelastic collision, momentum is conserved but KE is not — some energy goes to heat, sound, or deformation. A perfectly inelastic collision (objects stick together) loses the maximum possible KE. See the momentum calculator.

Are any real-world collisions perfectly elastic?

Almost none. Billiard balls are close (~95% KE conserved). Atomic and molecular collisions can be perfectly elastic. Car crashes are highly inelastic (crumple zones deliberately absorb KE as deformation). The elastic model is a useful idealization for problems where energy loss is negligible.

What if both objects are moving before the collision?

Use the general elastic collision formulas: v₁' = [(m₁−m₂)v₁ + 2m₂v₂]/(m₁+m₂) and v₂' = [(m₂−m₁)v₂ + 2m₁v₁]/(m₁+m₂). This calculator handles both the simple (one stationary) and general (both moving) cases.