3.3 Elastic and Inelastic Collisions | Translatory Motion | Physics 11

A collision is an interaction between two or more objects in which they exert forces on each other for a short time. Collisions are classified based on whether kinetic energy is conserved.

Elastic Collision

An elastic collision is one in which both the total linear momentum and the total kinetic energy of the system are conserved. No kinetic energy is lost to heat, sound, or deformation.

Perfectly elastic collisions are an idealization, but collisions between billiard balls, steel ball bearings, and subatomic particles are very close to perfectly elastic.

Conservation Laws for Elastic Collisions

For an elastic collision between two masses $m_{1}$ and $m_{2}$ with initial velocities $u_{1}$ and $u_{2}$ and final velocities $v_{1}$ and $v_{2}$ :

Conservation of Linear Momentum: $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$

Conservation of Kinetic Energy: $\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}$

Derivation of Final Velocities

By algebraically manipulating the two conservation equations, the key intermediate result is that the relative speed of approach equals the relative speed of separation:

$u_{1} - u_{2} = - (v_{1} - v_{2}) = v_{2} - v_{1}$

Using this with the momentum equation gives the final velocities:

Final Velocity of $m_{1}$ : $v_{1} = (\frac{m _{1} - m _{2}}{m _{1} + m _{2}}) u_{1} + (\frac{2 m _{2}}{m _{1} + m _{2}}) u_{2}$

Final Velocity of $m_{2}$ : $v_{2} = (\frac{2 m _{1}}{m _{1} + m _{2}}) u_{1} + (\frac{m _{2} - m _{1}}{m _{1} + m _{2}}) u_{2}$

Special Cases of Elastic Collisions

Case	Conditions	Final Velocities	Outcome
Equal masses	$m_{1} = m_{2}$	$v_{1} = u_{2}$ , $v_{2} = u_{1}$	Velocities exchanged
Equal masses, target at rest	$m_{1} = m_{2}$ , $u_{2} = 0$	$v_{1} = 0$ , $v_{2} = u_{1}$	First stops, second moves off
Light hits massive (at rest)	$m_{1} ≪ m_{2}$ , $u_{2} = 0$	$v_{1} \approx - u_{1}$ , $v_{2} \approx 0$	Light body bounces back
Massive hits light (at rest)	$m_{1} ≫ m_{2}$ , $u_{2} = 0$	$v_{1} \approx u_{1}$ , $v_{2} \approx 2 u_{1}$	Light body propelled at $2 u_{1}$

An inelastic collision is one in which total linear momentum is conserved, but total kinetic energy is not conserved. Some kinetic energy is converted into other forms such as heat, sound, or deformation energy.

Key principle (SLO P-11-B-13): In any closed (isolated) system, total momentum is always conserved regardless of the type of collision. However, kinetic energy may decrease because it can be transformed into internal energy (heat, deformation). Total energy is still conserved — it is only kinetic energy that changes form.

Perfectly Inelastic Collision

A perfectly inelastic collision is the extreme case where the two objects stick together after impact and move with a common velocity $V$ . Kinetic energy loss is maximum in this case.

Applying conservation of momentum: $m_{1} u_{1} + m_{2} u_{2} = (m_{1} + m_{2}) V$ $V = \frac{m _{1} u _{1} + m _{2} u _{2}}{m _{1} + m _{2}}$

Kinetic energy lost: $Δ K E = (\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2}) - \frac{1}{2} (m_{1} + m_{2}) V^{2}$

This lost kinetic energy appears as heat, sound, or permanent deformation.

Why Momentum is Always Conserved but KE May Not Be

Momentum conservation follows directly from Newton's Third Law: the internal forces between colliding objects are equal and opposite, so they cancel out and the total momentum of the system does not change. Kinetic energy, however, can be converted to other energy forms (heat, sound, deformation) through internal forces, so it is not necessarily conserved.

Comparison: Elastic vs Inelastic Collisions

Property	Elastic	Inelastic	Perfectly Inelastic
Momentum conserved?	✓ Yes	✓ Yes	✓ Yes
Kinetic energy conserved?	✓ Yes	✗ No (partial loss)	✗ No (maximum loss)
Objects stick together?	No	No	Yes
Example	Billiard balls	Car crash	Clay balls colliding

Reference

Elastic and Inelastic Collisions Explained

A collision is an interaction between two or more objects in which they exert forces on each other for a short time. Collisions are classified based on whether kinetic energy is conserved.

Elastic Collision

An elastic collision is one in which both the total linear momentum and the total kinetic energy of the system are conserved. No kinetic energy is lost to heat, sound, or deformation.

Perfectly elastic collisions are an idealization, but collisions between billiard balls, steel ball bearings, and subatomic particles are very close to perfectly elastic.

Conservation Laws for Elastic Collisions

For an elastic collision between two masses $m_{1}$ and $m_{2}$ with initial velocities $u_{1}$ and $u_{2}$ and final velocities $v_{1}$ and $v_{2}$ :

Conservation of Linear Momentum: $m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$

Conservation of Kinetic Energy: $\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2} = \frac{1}{2} m_{1} v_{1}^{2} + \frac{1}{2} m_{2} v_{2}^{2}$

Derivation of Final Velocities

By algebraically manipulating the two conservation equations, the key intermediate result is that the relative speed of approach equals the relative speed of separation:

$u_{1} - u_{2} = - (v_{1} - v_{2}) = v_{2} - v_{1}$

Using this with the momentum equation gives the final velocities:

Final Velocity of $m_{1}$ : $v_{1} = (\frac{m _{1} - m _{2}}{m _{1} + m _{2}}) u_{1} + (\frac{2 m _{2}}{m _{1} + m _{2}}) u_{2}$

Final Velocity of $m_{2}$ : $v_{2} = (\frac{2 m _{1}}{m _{1} + m _{2}}) u_{1} + (\frac{m _{2} - m _{1}}{m _{1} + m _{2}}) u_{2}$

Special Cases of Elastic Collisions

Case	Conditions	Final Velocities	Outcome
Equal masses	$m_{1} = m_{2}$	$v_{1} = u_{2}$ , $v_{2} = u_{1}$	Velocities exchanged
Equal masses, target at rest	$m_{1} = m_{2}$ , $u_{2} = 0$	$v_{1} = 0$ , $v_{2} = u_{1}$	First stops, second moves off
Light hits massive (at rest)	$m_{1} ≪ m_{2}$ , $u_{2} = 0$	$v_{1} \approx - u_{1}$ , $v_{2} \approx 0$	Light body bounces back
Massive hits light (at rest)	$m_{1} ≫ m_{2}$ , $u_{2} = 0$	$v_{1} \approx u_{1}$ , $v_{2} \approx 2 u_{1}$	Light body propelled at $2 u_{1}$

Inelastic Collision

Perfectly Inelastic Collision

A perfectly inelastic collision is the extreme case where the two objects stick together after impact and move with a common velocity $V$ . Kinetic energy loss is maximum in this case.

Applying conservation of momentum: $m_{1} u_{1} + m_{2} u_{2} = (m_{1} + m_{2}) V$ $V = \frac{m _{1} u _{1} + m _{2} u _{2}}{m _{1} + m _{2}}$

Kinetic energy lost: $Δ K E = (\frac{1}{2} m_{1} u_{1}^{2} + \frac{1}{2} m_{2} u_{2}^{2}) - \frac{1}{2} (m_{1} + m_{2}) V^{2}$

This lost kinetic energy appears as heat, sound, or permanent deformation.

Why Momentum is Always Conserved but KE May Not Be

Comparison: Elastic vs Inelastic Collisions

Property	Elastic	Inelastic	Perfectly Inelastic
Momentum conserved?	✓ Yes	✓ Yes	✓ Yes
Kinetic energy conserved?	✓ Yes	✗ No (partial loss)	✗ No (maximum loss)
Objects stick together?	No	No	Yes
Example	Billiard balls	Car crash	Clay balls colliding

Reference

Elastic and Inelastic Collisions Explained