In a one-dimensional collision or explosion, the objects move along a single straight line. If the net external impulse
during the interaction is negligible, the total momentum of the system is conserved.
Conservation of momentum in 1D
Choose one direction as positive. In this calculator, positive velocity means motion to the right and negative velocity
means motion to the left.
Main conservation equation.
\[
\begin{aligned}
m_1v_{1i}+m_2v_{2i} &= m_1v_{1f}+m_2v_{2f}
\end{aligned}
\]
The left side is the total momentum before the interaction, and the right side is the total momentum after the interaction.
The equation works for collisions, sticking-together impacts, rebounds, and explosions, as long as external impulse is negligible.
Perfectly inelastic collision
In a perfectly inelastic collision, the objects stick together and move with a common final velocity \(v_f\).
Common final velocity.
\[
\begin{aligned}
m_1v_{1i}+m_2v_{2i} &= (m_1+m_2)v_f \\
v_f &= \frac{m_1v_{1i}+m_2v_{2i}}{m_1+m_2}
\end{aligned}
\]
Momentum is conserved, but kinetic energy usually decreases because some energy becomes heat, sound, deformation, or internal energy.
Elastic collision
In an elastic collision, both momentum and kinetic energy are conserved. For two objects in one dimension, the final velocities are:
Elastic-collision formulas.
\[
\begin{aligned}
v_{1f} &= \frac{(m_1-m_2)v_{1i}+2m_2v_{2i}}{m_1+m_2} \\
v_{2f} &= \frac{2m_1v_{1i}+(m_2-m_1)v_{2i}}{m_1+m_2}
\end{aligned}
\]
These equations are useful when the coefficient of restitution is \(e=1\).
Coefficient of restitution
The coefficient of restitution describes how much relative speed remains after the collision. It ranges from \(0\) to \(1\)
for ordinary collisions.
Restitution definition.
\[
\begin{aligned}
e &= \frac{v_{2f}-v_{1f}}{v_{1i}-v_{2i}}
\end{aligned}
\]
Combining this with momentum conservation gives:
Final velocities using restitution.
\[
\begin{aligned}
v_{1f} &= \frac{m_1v_{1i}+m_2v_{2i}-m_2e(v_{1i}-v_{2i})}{m_1+m_2} \\
v_{2f} &= \frac{m_1v_{1i}+m_2v_{2i}+m_1e(v_{1i}-v_{2i})}{m_1+m_2}
\end{aligned}
\]
When \(e=1\), this becomes an elastic collision. When \(e=0\), the relative final speed is zero, so the objects leave with
the same velocity.
Explosions and separation
An explosion is also handled by conservation of momentum. The difference is that kinetic energy can increase because internal
energy is released. For example, if two objects are initially at rest together, their final momenta must be equal in size and
opposite in direction:
Explosion from rest.
\[
\begin{aligned}
0 &= m_1v_{1f}+m_2v_{2f}
\end{aligned}
\]
If one final velocity is known, the other can be found directly from the momentum equation.
Solving for unknown velocities or masses
Momentum conservation can be rearranged to solve for a missing final velocity or a missing mass.
Unknown final velocity 1.
\[
\begin{aligned}
v_{1f} &= \frac{m_1v_{1i}+m_2v_{2i}-m_2v_{2f}}{m_1}
\end{aligned}
\]
Unknown final velocity 2.
\[
\begin{aligned}
v_{2f} &= \frac{m_1v_{1i}+m_2v_{2i}-m_1v_{1f}}{m_2}
\end{aligned}
\]
Unknown mass 1.
\[
\begin{aligned}
m_1 &= \frac{m_2(v_{2f}-v_{2i})}{v_{1i}-v_{1f}}
\end{aligned}
\]
Unknown mass 2.
\[
\begin{aligned}
m_2 &= \frac{m_1(v_{1f}-v_{1i})}{v_{2i}-v_{2f}}
\end{aligned}
\]
Worked example
Suppose \(m_1=2\ \mathrm{kg}\), \(v_{1i}=10\ \mathrm{m\,s^{-1}}\), \(m_2=3\ \mathrm{kg}\), and
\(v_{2i}=0\). If the objects stick together, then:
Compute the common final velocity.
\[
\begin{aligned}
v_f &= \frac{m_1v_{1i}+m_2v_{2i}}{m_1+m_2} \\
&= \frac{(2)(10)+(3)(0)}{2+3} \\
&= 4\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
The final velocity is positive, so the stuck-together objects move to the right.
Formula summary
| Case |
Formula |
Meaning |
| Momentum conservation |
\(m_1v_{1i}+m_2v_{2i}=m_1v_{1f}+m_2v_{2f}\) |
Total momentum before equals total momentum after |
| Perfectly inelastic |
\(v_f=(m_1v_{1i}+m_2v_{2i})/(m_1+m_2)\) |
Objects stick together |
| Restitution |
\(e=(v_{2f}-v_{1f})/(v_{1i}-v_{2i})\) |
Controls how elastic the collision is |
| Initial kinetic energy |
\(K_i=\tfrac12m_1v_{1i}^2+\tfrac12m_2v_{2i}^2\) |
Total kinetic energy before |
| Final kinetic energy |
\(K_f=\tfrac12m_1v_{1f}^2+\tfrac12m_2v_{2f}^2\) |
Total kinetic energy after |
The signs matter. A negative final velocity means that object moves left after the interaction; a positive final velocity means it moves right.
