3.3.4 Elastic Collision

Consider a head-on elastic 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.

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

Since this is an elastic collision, we can form the relative speed equation.

$\displaystyle \begin{aligned}\text{RSOA}&=\text{RSOS}\\{{u}_{1}}-{{u}_{2}}&={{v}_{2}}-{{v}_{1}}\end{aligned}$

Substituting $\displaystyle {{v}_{2}}={{u}_{1}}-{{u}_{2}}+{{v}_{1}}$ into the PCOM equation, we obtain

$\displaystyle {{v}_{1}}=\frac{{{{m}_{1}}-{{m}_{2}}}}{{{{m}_{1}}+{{m}_{2}}}}{{u}_{1}}+\frac{{2{{m}_{2}}}}{{{{m}_{1}}+{{m}_{2}}}}{{u}_{2}}$

Similarly, substituting $\displaystyle {{v}_{1}}={{u}_{2}}-{{u}_{1}}+{{v}_{2}}$ into the PCOM equation, we obtain

$\displaystyle {{v}_{2}}=\frac{{2{{m}_{2}}}}{{{{m}_{1}}+{{m}_{2}}}}{{u}_{1}}+\frac{{{{m}_{2}}-{{m}_{1}}}}{{{{m}_{1}}+{{m}_{2}}}}{{u}_{2}}$

There is no need to memorize these formulae. Nobody does. However, the outcomes of certain “classic” collisions are worth remembering.

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Elastic “Jousting” Example

What is the outcome of a head-on elastic collision between two masses of equal masses m, both moving at the same speed u in opposite directions?

Elastic: $\displaystyle \begin{aligned}\text{RSOA}&=\text{RSOS}\\u-(-u)&={{v}_{2}}-{{v}_{1}}\\{{v}_{2}}-{{v}_{1}}&=2u\text{ }\cdots \cdots \text{(1)}\end{aligned}$

PCOM: $\displaystyle \begin{aligned}\sum {{p}_{i}}&=\sum {{p}_{f}}\\mu+m(-u)&=m{{v}_{1}}+m{{v}_{2}}\\{{v}_{1}}+{{v}_{2}}&=0\text{ }\cdots \cdots \text{(2)}\end{aligned}$

Solving (1) and (2) equations yield

$\displaystyle {{v}_{2}}=u$

$\displaystyle {{v}_{1}}=-u$

Since rightward is positive, the negative sign informs us that v₁ is actually leftward.

Let’s look at the graphs depicting the momentum and KE variations with time for this collision. For simplicity, we have assumed that the momentums change linearly during the collision (i.e. we have assumed a constant impact force during collision).

Notice that:

The two masses approach and separate from each other at relative speed of 2u.
The total momentum remains constant (at zero) throughout the collision.
The total KE drops to zero at one point during the collision but returns to 100% by the end of the collision. The KE is temporarily stored as some form of potential energy (perhaps as elastic PE as the balls become compressed during the collision), but is returned fully as KE eventually (as the balls uncompressed to their original state after the collision).

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Elastic “Sitting Duck” Example

What is the outcome of a head-on elastic collision between two masses of equal masses m, one with initial speed u, and the other initially at rest.

Elastic: $\displaystyle \begin{aligned}\text{RSOA}&=\text{RSOS}\\u-0&={{v}_{2}}-{{v}_{1}}\\{{v}_{2}}-{{v}_{1}}&=u\text{ }\cdots \cdots \text{(1)}\end{aligned}$

PCOM: $\displaystyle \begin{aligned}\sum {{p}_{i}}&=\sum {{p}_{f}}\\mu+0&=m{{v}_{1}}+m{{v}_{2}}\\{{v}_{1}}+{{v}_{2}}&=u\text{ }\cdots \cdots \text{(2)}\end{aligned}$

Solving (1) and (2) equations yield

$\displaystyle {{v}_{2}}=u$

$\displaystyle {{v}_{1}}=0$

The following graphs depicts the momentum and KE variations with time for this collision. For simplicity, we have assumed that the momentums change linearly during the collision (i.e. we have assumed a constant impact force during collision).

Notice that:

The two masses approach and separate from each other at relative speed of u.
The total momentum remains constant (at mu) throughout the collision.
The total KE drops to 50% at one point during the collision but returns to 100% by the end of the collision. Note that total KE cannot possibly drop to zero at any instant during the collision because that would imply both masses come to rest at the same instant, thus violating the PCOM for this collision.

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Demonstration

Gangnam Style

Animation

Classic Elastic Collisions

Concept Test

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