3.3.5 Perfectly Inelastic Collision

Consider a head-on perfectly inelastic collision between two masses of mass m₁ and m₂ with initial velocities u₁ and u₂. What are the velocities of the two masses v₁ and v₂ after the collision?

Since total momentum is conserved (in any collision, elastic or not), we can form the PCOM equation.

$\begin{aligned}\sum {{p}_{i}}&=\sum {{p}_{f}}\\{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}&={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}\end{aligned}$

Since this is a perfectly inelastic collision,

${{v}_{1}}={{v}_{2}}=v$

Replacing v₁ and v₂ in the PCOM equation with v,

$\begin{aligned}{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}&=({{m}_{1}}+{{m}_{2}})v\\v&=\frac{{{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}}}{{{{m}_{1}}+{{m}_{2}}}}\end{aligned}$

Perfectly Inelastic “Jousting” Example

What is the outcome of the collision depicted below? Perfectly inelastic collision of two equal masses m, both with initial speed of u.

PCOM:

$\begin{aligned}\sum {{p}_{i}}&=\sum {{p}_{f}}\\mu+m(-u)&=2mv\\v&=0\end{aligned}$

The following graphs depicts the momentum and KE variations for this collision.

Notice that:

For simplicity, we have assumed a constant contact force during the collision (so the momentums change linearly).
The total momentum remains constant (at zero) throughout the collision.
The total KE drops to 0% permanently after the collision. So what happened to all the KE? Stored as intermolecular potential energy if the chemical structure is permanently deformed. Dissipated as heat if the molecules vibrate. Carried away as sound energy. These are the usual suspects.

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Perfectly Inelastic “Sitting Duck” Example

What is the outcome of the collision depicted below? Perfectly inelastic collision of two equal masses m, one with initial speed of u and the other initially at rest.

PCOM:

$\begin{aligned}\sum {{p}_{i}}&=\sum {{p}_{f}}\\mu+0&=2mv\\v&=\frac{u}{2}\end{aligned}$

The following graphs depicts the momentum and KE variations for this collision.

Notice that:

For simplicity, we have assumed a constant contact force during the collision (so the momentums change linearly).
The total momentum remains constant (at mu) throughout the collision.
The total KE drops to 50% permanently during the collision.

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Animations

Classic Perrfectly Inelastic Collisions

Concept Test

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