In physics, a perfectly elastic collision is one where absolutely no kinetic energy is lost. Think of two perfectly hard billiard balls clicking together. Both momentum and kinetic energy are perfectly conserved, allowing us to predict exactly how fast they will bounce apart.
Mathematical Formula & Logic
Final Velocity (Object 1)
v1 = (u1 * (m1 - m2) + 2 * m2 * u2) / (m1 + m2)
- v1 = Final Velocity of Object 1 (m/s)
- u1 = Initial Velocity of Object 1 (m/s)
- u2 = Initial Velocity of Object 2 (m/s)
- m1 = Mass of Object 1 (kg)
- m2 = Mass of Object 2 (kg)
Final Velocity (Object 2)
v2 = (u2 * (m2 - m1) + 2 * m1 * u1) / (m1 + m2)
- v2 = Final Velocity of Object 2 (m/s)
- u1 = Initial Velocity of Object 1 (m/s)
- u2 = Initial Velocity of Object 2 (m/s)
- m1 = Mass of Object 1 (kg)
- m2 = Mass of Object 2 (kg)
Step-by-Step Example
If a 1kg ball moving at 5m/s hits a stationary 1kg ball, they swap velocities. The first ball stops (0m/s) and the second ball shoots forward at 5m/s.
If a massive truck (10,000kg) hits a stationary tiny pebble (1kg) at 10m/s, the truck keeps going at ~10m/s, but the pebble gets launched forward at nearly twice the speed (~20m/s).
Reference Data & Values
m1
m2
u1
u2
v1
v2
1
1
5
0
0
5
1
1
10
-5
-5
10
10000
1
10
0
9.999
19.998
1
10000
10
0
-9.999
0.00199
2
3
4
-2
-3.2
2.8
Frequently Asked Questions
An elastic collision is a collision where there is no loss of kinetic energy in the system. Both momentum and kinetic energy are conserved.
In an elastic collision, kinetic energy is conserved (objects bounce off each other perfectly). In an inelastic collision, some kinetic energy is lost to heat, sound, or deformation (e.g., objects dent or stick together).
No macroscopic collision is perfectly elastic because some energy is always lost to sound and heat. However, collisions between hard spheres (like Newton's Cradle or billiard balls) are very close to perfectly elastic.
If they hit head-on in 1D, they perfectly swap velocities. If one was stationary, the moving one stops dead, and the stationary one takes off at the exact speed the first one had.
The light object will bounce back with nearly the exact same speed in the opposite direction (velocity becomes negative). The heavy wall will barely move.
Yes! Momentum is conserved in ALL isolated collisions (both elastic and inelastic). It is only Kinetic Energy that is lost during inelastic collisions.
Because kinetic energy must be conserved. When a massive train hits a stationary pebble, the train doesn't slow down. To balance the kinetic energy equation, the tiny pebble must absorb energy by shooting forward at twice the train's speed.
It uses 1D vectors. Positive velocities mean moving right; negative velocities mean moving left. If a result is negative, the object is moving to the left after the crash.