A partially inelastic collision is a collision in which total momentum is conserved, but kinetic energy is not fully
conserved. The amount of bounce is described by the coefficient of restitution \(e\).
Sign convention
In one dimension, direction is represented by signs. In this calculator, positive velocity means motion to the right,
and negative velocity means motion to the left.
Momentum conservation
If the external impulse during the collision is negligible, total momentum is conserved:
Total momentum before equals total momentum after.
\[
\begin{aligned}
m_1u_1+m_2u_2 &= m_1v_1+m_2v_2
\end{aligned}
\]
Momentum conservation alone is not enough to solve both final velocities. The coefficient of restitution supplies the
second equation.
Coefficient of restitution
The coefficient of restitution compares the speed of separation after collision to the speed of approach before collision.
Restitution definition.
\[
\begin{aligned}
e &= \frac{v_2-v_1}{u_1-u_2}
\end{aligned}
\]
Rearranged:
\[
\begin{aligned}
v_2-v_1 &= e(u_1-u_2)
\end{aligned}
\]
The value of \(e\) determines the collision type:
- \(e=0\): perfectly inelastic collision; the objects leave with the same velocity.
- \(0<e<1\): partially inelastic collision; the objects bounce, but kinetic energy decreases.
- \(e=1\): perfectly elastic collision; kinetic energy is conserved.
Final velocity formulas
Combining momentum conservation with the restitution equation gives:
Final velocity of object 1.
\[
\begin{aligned}
v_1
&=
\frac{m_1u_1+m_2u_2-m_2e(u_1-u_2)}{m_1+m_2}
\end{aligned}
\]
Final velocity of object 2.
\[
\begin{aligned}
v_2
&=
\frac{m_1u_1+m_2u_2+m_1e(u_1-u_2)}{m_1+m_2}
\end{aligned}
\]
Kinetic energy loss
The initial and final kinetic energies are:
\[
\begin{aligned}
K_i &= \frac12m_1u_1^2+\frac12m_2u_2^2\\
K_f &= \frac12m_1v_1^2+\frac12m_2v_2^2
\end{aligned}
\]
The kinetic energy transformed into heat, sound, deformation, and internal energy is:
Energy lost.
\[
\begin{aligned}
K_{\mathrm{lost}} &= K_i-K_f
\end{aligned}
\]
A compact formula for the kinetic energy lost in a one-dimensional collision with coefficient \(e\) is:
Reduced-mass loss formula.
\[
\begin{aligned}
K_{\mathrm{lost}}
&=
\frac12\mu(1-e^2)(u_1-u_2)^2,
\qquad
\mu=\frac{m_1m_2}{m_1+m_2}
\end{aligned}
\]
This formula shows why the energy loss is zero when \(e=1\), and largest when \(e=0\).
Perfectly inelastic limit
When \(e=0\), the restitution equation becomes:
\[
\begin{aligned}
v_2-v_1 &= 0
\end{aligned}
\]
Therefore \(v_1=v_2=v_f\). The common final velocity is:
\[
\begin{aligned}
v_f
&=
\frac{m_1u_1+m_2u_2}{m_1+m_2}
\end{aligned}
\]
Elastic limit
When \(e=1\), the speed of approach equals the speed of separation:
\[
\begin{aligned}
u_1-u_2 &= v_2-v_1
\end{aligned}
\]
This is the perfectly elastic case, where total kinetic energy is conserved.
Worked example
Suppose \(e=0.6\), \(m_1=3\ \mathrm{kg}\), \(u_1=8\ \mathrm{m\,s^{-1}}\),
\(m_2=2\ \mathrm{kg}\), and \(u_2=0\).
Final velocity of object 1.
\[
\begin{aligned}
v_1
&=
\frac{(3)(8)+(2)(0)-(2)(0.6)(8-0)}{3+2}\\
&=
\frac{24-9.6}{5}\\
&=
2.88\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
Final velocity of object 2.
\[
\begin{aligned}
v_2
&=
\frac{(3)(8)+(2)(0)+(3)(0.6)(8-0)}{3+2}\\
&=
\frac{24+14.4}{5}\\
&=
7.68\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
Energy loss.
\[
\begin{aligned}
K_i &= \frac12(3)(8)^2+\frac12(2)(0)^2=96\ \mathrm{J}\\
K_f &= \frac12(3)(2.88)^2+\frac12(2)(7.68)^2\approx71.42\ \mathrm{J}\\
K_{\mathrm{lost}}&\approx24.58\ \mathrm{J}
\end{aligned}
\]
Formula summary
| Concept |
Formula |
Meaning |
| Momentum conservation |
\(m_1u_1+m_2u_2=m_1v_1+m_2v_2\) |
Total momentum is conserved when external impulse is negligible |
| Coefficient of restitution |
\(e=\frac{v_2-v_1}{u_1-u_2}\) |
Ratio of separation speed to approach speed |
| Final velocity 1 |
\(v_1=\frac{m_1u_1+m_2u_2-m_2e(u_1-u_2)}{m_1+m_2}\) |
Velocity of object 1 after collision |
| Final velocity 2 |
\(v_2=\frac{m_1u_1+m_2u_2+m_1e(u_1-u_2)}{m_1+m_2}\) |
Velocity of object 2 after collision |
| Kinetic energy lost |
\(K_{\mathrm{lost}}=K_i-K_f\) |
Energy transformed into non-mechanical forms |
| Reduced-mass loss formula |
\(K_{\mathrm{lost}}=\frac12\mu(1-e^2)(u_1-u_2)^2\) |
Shows how \(e\) controls kinetic energy loss |
| Reduced mass |
\(\mu=\frac{m_1m_2}{m_1+m_2}\) |
Effective mass for relative motion |
Lower \(e\) means less bounce and more kinetic energy lost. Higher \(e\) means more bounce and less energy lost.
