Momentum measures how much motion an object has. For one-dimensional motion, momentum is the product of mass and velocity.
Because velocity has direction, momentum also has direction.
Momentum
Linear momentum.
\[
\begin{aligned}
p &= mv
\end{aligned}
\]
If an object changes velocity from \(v_i\) to \(v_f\), its change in momentum is:
Change in momentum.
\[
\begin{aligned}
\Delta p &= p_f-p_i \\
&= mv_f-mv_i \\
&= m(v_f-v_i) \\
&= m\Delta v
\end{aligned}
\]
Impulse-momentum theorem
Impulse is the effect of a force acting over a time interval. The impulse-momentum theorem states that impulse equals
change in momentum.
Impulse-momentum theorem.
\[
\begin{aligned}
J &= \Delta p
\end{aligned}
\]
If the force is approximately constant, or if an average force is used for a variable force, impulse is:
Average-force impulse.
\[
\begin{aligned}
J &= F_{\mathrm{avg}}\Delta t
\end{aligned}
\]
Therefore, the main working equation is:
Main equation.
\[
\begin{aligned}
F_{\mathrm{avg}}\Delta t &= m\Delta v
\end{aligned}
\]
Solving for different quantities
The impulse-momentum equation can be rearranged depending on which variable is unknown.
Average force.
\[
\begin{aligned}
F_{\mathrm{avg}} &= \frac{\Delta p}{\Delta t}
\end{aligned}
\]
Contact time.
\[
\begin{aligned}
\Delta t &= \frac{\Delta p}{F_{\mathrm{avg}}}
\end{aligned}
\]
Mass.
\[
\begin{aligned}
m &= \frac{\Delta p}{\Delta v}
\end{aligned}
\]
Velocity change.
\[
\begin{aligned}
\Delta v &= \frac{\Delta p}{m}
\end{aligned}
\]
Variable force and graph area
Real collision forces are often not constant. For example, a bat-ball impact or car crash may have a force that rises to
a peak and then falls. The impulse is still the area under the force-time graph:
Impulse as force-time area.
\[
\begin{aligned}
J &= \int_{t_i}^{t_f} F(t)\,dt
\end{aligned}
\]
The average force is the constant force that would give the same area over the same time interval:
Average force from area.
\[
\begin{aligned}
F_{\mathrm{avg}} &= \frac{1}{\Delta t}\int_{t_i}^{t_f} F(t)\,dt
\end{aligned}
\]
Worked example
A baseball of mass \(0.15\ \mathrm{kg}\) changes velocity from \(0\ \mathrm{m\,s^{-1}}\) to
\(48\ \mathrm{m\,s^{-1}}\) during a \(0.002\ \mathrm{s}\) contact time.
Momentum change.
\[
\begin{aligned}
\Delta p &= m(v_f-v_i) \\
&= 0.15(48-0) \\
&= 7.2\ \mathrm{kg\,m\,s^{-1}}
\end{aligned}
\]
Average force.
\[
\begin{aligned}
F_{\mathrm{avg}} &= \frac{\Delta p}{\Delta t} \\
&= \frac{7.2}{0.002} \\
&= 3600\ \mathrm{N}
\end{aligned}
\]
Formula summary
| Goal |
Formula |
Meaning |
| Momentum |
\(p=mv\) |
Mass times velocity |
| Change in momentum |
\(\Delta p=m(v_f-v_i)\) |
Final momentum minus initial momentum |
| Impulse |
\(J=\Delta p\) |
Impulse equals change in momentum |
| Average force |
\(F_{\mathrm{avg}}=\Delta p/\Delta t\) |
Force that gives the same impulse over the same time |
| Force-time area |
\(J=\int F(t)\,dt\) |
Area under the force-time graph |
This calculator uses one-dimensional motion. For vector momentum problems in two or three dimensions, the same theorem
applies component by component.
