Mechanicsp = mv

Collisions Simulator

Simulate 1D elastic and inelastic collisions. Adjust masses and initial velocities to explore momentum and kinetic energy conservation.

Parameters

Collision type

Mass m₁

Velocity v₁

m/s

Mass m₂

Velocity v₂

m/s

p₁0.000 kg·m/s

p₂0.000 kg·m/s

p_total0.000 kg·m/s

v₁ after0.000 m/s

v₂ after0.000 m/s

KE before0.000 J

KE after0.000 J

ΔKE0.000 J

About Collisions

A collision is an event in which two or more objects exert forces on each other over a short time interval. The total linear momentum of an isolated system is always conserved in any collision. Kinetic energy, however, is only conserved in perfectly elastic collisions.

ℹMomentum is always conserved in collisions (assuming no external forces). Kinetic energy is conserved only in elastic collisions — in inelastic collisions, some KE converts to heat, sound, or deformation.

Key Variables

Symbol	Name	Unit	Description
m₁, m₂	Masses	kg	Masses of the two blocks
v₁, v₂	Initial Velocities	m/s	Velocities before collision (positive = rightward)
v₁f, v₂f	Final Velocities	m/s	Velocities after collision
p₁, p₂	Momenta	kg·m/s	p = mv for each block
p_total	Total Momentum	kg·m/s	p₁ + p₂ — conserved in all collisions
KE_before	Initial KE	J	½m₁v₁² + ½m₂v₂²
KE_after	Final KE	J	½m₁v₁f² + ½m₂v₂f²
ΔKE	Energy Lost	J	KE_before − KE_after (zero for elastic)

Elastic Collision Example

m₁ = 2 kg, v₁ = 3 m/s, m₂ = 1 kg, v₂ = −1 m/s (elastic collision):

v_{1 f} = \frac{( m _{1} - m _{2} ) v _{1} + 2 m _{2} v _{2}}{m _{1} + m _{2}} = \frac{( 2 - 1 ) ( 3 ) + 2 ( 1 ) ( - 1 )}{3} = \frac{1}{3} \approx 0.33 m/s

v_{2 f} = \frac{( m _{2} - m _{1} ) v _{2} + 2 m _{1} v _{1}}{m _{1} + m _{2}} = \frac{( 1 - 2 ) ( - 1 ) + 2 ( 2 ) ( 3 )}{3} = \frac{13}{3} \approx 4.33 m/s

Inelastic (Perfectly Inelastic) Collision

In a perfectly inelastic collision, the two objects stick together and move as one. This represents maximum kinetic energy loss while still conserving momentum.

v_{f} = \frac{m _{1} v _{1} + m _{2} v _{2}}{m _{1} + m _{2}}

💡Switch between Elastic and Inelastic modes in the simulator. Notice total momentum is always conserved, but the KE_after readout changes significantly between modes.

Special Cases

Equal masses (elastic): objects exchange velocities — classic Newton's Cradle behaviour.
Very heavy object hits stationary light object (elastic): heavy barely slows, light bounces at ~2× heavy's speed.
Light hits very heavy stationary object (elastic): light bounces back at nearly the same speed.
Head-on collision with equal speeds (elastic): both reverse directions.

Collision Formulas

Conservation of Momentum

m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1 f} + m_{2} v_{2 f}

Elastic Collision Velocities

v_{1 f} = \frac{( m _{1} - m _{2} ) v _{1} + 2 m _{2} v _{2}}{m _{1} + m _{2}} v_{2 f} = \frac{( m _{2} - m _{1} ) v _{2} + 2 m _{1} v _{1}}{m _{1} + m _{2}}

Perfectly Inelastic Collision Velocity

v_{f} = \frac{m _{1} v _{1} + m _{2} v _{2}}{m _{1} + m _{2}}

Kinetic Energy Before and After

K E_{before} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}

K E_{after} = \frac{1}{2} m_{1} v_{1 f}^{2} + \frac{1}{2} m_{2} v_{2 f}^{2}

Quantity	Elastic	Inelastic
Momentum	Conserved	Conserved
Kinetic Energy	Conserved	NOT conserved (ΔKE lost as heat/sound)
Final velocities	Two separate	One shared velocity
Objects	Bounce apart	Stick together

ℹIn elastic collisions, both momentum AND kinetic energy are conserved simultaneously. This gives two equations for two unknowns (v1f, v2f), yielding the formulas above.

Frequently Asked Questions

Is momentum always conserved in a collision?

Yes, if there are no external forces on the system. The internal collision forces are equal and opposite (Newton's Third Law), so they cancel when summing over the whole system. Momentum is always conserved in both elastic and inelastic collisions.

What is the coefficient of restitution?

The coefficient of restitution e = |v2f − v1f| / |v1 − v2|. For perfectly elastic collisions e = 1, for perfectly inelastic e = 0. Values between 0 and 1 represent partially inelastic collisions. This simulator shows the two extreme cases.

What happens when equal masses collide elastically?

The velocities are exchanged. If m₁ = m₂ = m, then v1f = v2 and v2f = v1. This is why in Newton's Cradle, one ball hitting a stationary ball sends exactly one ball out the other side.

Can I set a negative velocity?

Yes. Negative velocity means the block moves to the left. This allows head-on collisions where both blocks approach each other, or scenarios where one block overtakes another from behind.

What causes kinetic energy loss in an inelastic collision?

The 'lost' KE is converted to other forms of energy: heat from deformation, sound (the collision 'thud'), permanent deformation of materials, and vibrations. Total energy including these forms is still conserved — only mechanical KE decreases.