Momentum and impulse in one dimension are based on a signed direction. In this calculator, positive velocity,
positive impulse, and positive force point to the right. Negative values point to the left.
Impulse-momentum theorem
Impulse is the area under a force-time graph, and it equals the change in momentum:
Main theorem.
\[
\begin{aligned}
J &= \int_{t_0}^{t_1}F(t)\,dt \\
&= \Delta p \\
&= m(v-u)
\end{aligned}
\]
If the force varies during an impact, the average force is the constant force that would produce the same impulse
during the same contact time:
Average force relation.
\[
\begin{aligned}
J &= F_{\mathrm{avg}}\Delta t
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
F_{\mathrm{avg}} &= \frac{J}{\Delta t}, \qquad
\Delta t = \frac{J}{F_{\mathrm{avg}}}
\end{aligned}
\]
Final velocity from impulse
If the mass and initial velocity are known, impulse gives the final velocity:
Final velocity formula.
\[
\begin{aligned}
v &= u+\frac{J}{m}
\end{aligned}
\]
A positive impulse increases rightward momentum. A negative impulse reduces rightward momentum or can reverse the object.
Collision force and contact time
For a fixed impulse, increasing the contact time reduces the average force:
\[
\begin{aligned}
F_{\mathrm{avg}}=\frac{\Delta p}{\Delta t}
\end{aligned}
\]
This is why airbags, padding, crumple zones, and sports follow-through reduce peak and average impact forces: they increase
the time over which momentum changes.
Two-body collisions with external impulse
In a two-object collision, the internal contact forces change the individual momenta. If there is also an external impulse
on the whole system, the system momentum changes by that external impulse:
System momentum balance.
\[
\begin{aligned}
m_1v_1+m_2v_2
&=
m_1u_1+m_2u_2+J_{\mathrm{ext}}
\end{aligned}
\]
If \(J_{\mathrm{ext}}=0\), the system is closed during the collision and total momentum is conserved.
Restitution model in 1D
A coefficient of restitution \(e\) describes how much relative speed remains after the collision:
Restitution equation.
\[
\begin{aligned}
v_2-v_1 &= e(u_1-u_2)
\end{aligned}
\]
Combining restitution with the momentum equation gives:
Final velocities with external impulse.
\[
\begin{aligned}
v_1 &=
\frac{m_1u_1+m_2u_2+J_{\mathrm{ext}}-m_2e(u_1-u_2)}{m_1+m_2} \\[3pt]
v_2 &=
\frac{m_1u_1+m_2u_2+J_{\mathrm{ext}}+m_1e(u_1-u_2)}{m_1+m_2}
\end{aligned}
\]
Special cases:
- \(e=1\): elastic collision.
- \(e=0\): perfectly inelastic collision, so both objects share the same final velocity.
- \(0<e<1\): partially elastic collision.
Impulse on each object
After the final velocities are known, the impulse on each body is:
\[
\begin{aligned}
J_1 &= m_1(v_1-u_1) \\
J_2 &= m_2(v_2-u_2)
\end{aligned}
\]
If there is no external impulse, \(J_1+J_2=0\). This means the objects exchange equal and opposite impulses.
If an external impulse acts, then \(J_1+J_2=J_{\mathrm{ext}}\).
Worked example: crash force
A \(1200\ \mathrm{kg}\) car changes velocity from \(12\ \mathrm{m\,s^{-1}}\) to \(0\ \mathrm{m\,s^{-1}}\)
in \(0.15\ \mathrm{s}\).
Compute impulse.
\[
\begin{aligned}
J &= m(v-u)\\
&= (1200)(0-12)\\
&= -14400\ \mathrm{N\,s}
\end{aligned}
\]
Compute average force.
\[
\begin{aligned}
F_{\mathrm{avg}} &= \frac{J}{\Delta t}\\
&= \frac{-14400}{0.15}\\
&= -96000\ \mathrm{N}
\end{aligned}
\]
The negative sign means the force acts opposite the original positive direction of motion.
Formula summary
| Use case |
Formula |
Meaning |
| Impulse-momentum theorem |
\(J=\Delta p=m(v-u)\) |
Impulse equals change in momentum |
| Average force |
\(F_{\mathrm{avg}}=J/\Delta t\) |
Average impact force over the contact time |
| Contact time |
\(\Delta t=J/F_{\mathrm{avg}}\) |
Time needed for a given impulse and average force |
| Final velocity |
\(v=u+J/m\) |
Velocity after the impulse |
| System momentum with external impulse |
\(m_1v_1+m_2v_2=m_1u_1+m_2u_2+J_{\mathrm{ext}}\) |
External impulse changes total system momentum |
| Restitution |
\(v_2-v_1=e(u_1-u_2)\) |
Models bounce in a 1D collision |
The shaded force-time area in the calculator is impulse. For the same momentum change, larger contact time means smaller average force.
