A ballistic pendulum measures the speed of a projectile by letting the projectile embed in a suspended block.
The analysis has two separate stages: a short perfectly inelastic collision, followed by a pendulum swing.
Stage 1: perfectly inelastic collision
During the short impact, the projectile sticks in the block. Mechanical energy is not conserved during this impact,
but horizontal momentum is approximately conserved because the external impulse during the collision is small.
Momentum conservation during impact.
\[
\begin{aligned}
m_p v_0 &= (m_p+m_b)v'
\end{aligned}
\]
Here \(m_p\) is the projectile mass, \(m_b\) is the block mass, \(v_0\) is the projectile speed before impact,
and \(v'\) is the speed of the combined block-projectile system immediately after impact.
Projectile speed formula
Solving the momentum equation for the projectile speed gives:
Projectile speed.
\[
\begin{aligned}
v_0 &= \frac{m_p+m_b}{m_p}v'
\end{aligned}
\]
Projectile mass formula
If the projectile speed is known and the projectile mass is unknown, rearrange the same momentum equation:
Projectile mass.
\[
\begin{aligned}
m_pv_0 &= (m_p+m_b)v'\\
m_p(v_0-v') &= m_bv'\\
m_p &= \frac{m_bv'}{v_0-v'}
\end{aligned}
\]
This requires \(v_0>v'\), because the projectile must be faster before impact than the combined pendulum is after impact.
Stage 2: pendulum swing
After the impact, the projectile and block move together. During the swing, mechanical energy is conserved if air resistance
and pivot friction are negligible.
Energy conservation during swing.
\[
\begin{aligned}
\frac12(m_p+m_b)v'^2 &= (m_p+m_b)gh
\end{aligned}
\]
The combined mass cancels:
\[
\begin{aligned}
v' &= \sqrt{2gh}
\end{aligned}
\]
Height from swing angle
If the vertical rise height is not measured directly, it can be computed from the pendulum length and maximum swing angle.
Pendulum geometry.
\[
\begin{aligned}
h &= L(1-\cos\theta)
\end{aligned}
\]
Here \(L\) is the pendulum length and \(\theta\) is the maximum angle measured from the vertical.
Energy loss during impact
The ballistic pendulum collision is perfectly inelastic, so kinetic energy decreases during the impact. The lost kinetic
energy becomes heat, sound, deformation, and internal energy.
Impact energy loss.
\[
\begin{aligned}
K_{\mathrm{lost}}
&=
\frac12m_pv_0^2-\frac12(m_p+m_b)v'^2
\end{aligned}
\]
This energy loss happens during the collision stage, not during the ideal pendulum swing stage.
Complete projectile speed formula
Combining \(v'=\sqrt{2gh}\) with the momentum equation gives:
\[
\begin{aligned}
v_0
&=
\frac{m_p+m_b}{m_p}\sqrt{2gh}
\end{aligned}
\]
If the height is found from angle:
\[
\begin{aligned}
v_0
&=
\frac{m_p+m_b}{m_p}
\sqrt{2gL(1-\cos\theta)}
\end{aligned}
\]
Worked example
Suppose \(m_p=0.01\ \mathrm{kg}\), \(m_b=1.5\ \mathrm{kg}\), \(h=0.12\ \mathrm{m}\), and
\(g=9.80665\ \mathrm{m\,s^{-2}}\).
Find speed just after impact.
\[
\begin{aligned}
v' &= \sqrt{2gh}\\
&= \sqrt{2(9.80665)(0.12)}\\
&\approx 1.53\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
Find projectile speed before impact.
\[
\begin{aligned}
v_0
&=
\frac{m_p+m_b}{m_p}v'\\
&=
\frac{0.01+1.5}{0.01}(1.53)\\
&\approx 232\ \mathrm{m\,s^{-1}}
\end{aligned}
\]
A result near \(380\ \mathrm{m\,s^{-1}}\) with these same masses would require a larger rise height, about
\(0.32\ \mathrm{m}\).
Formula summary
| Stage |
Formula |
Meaning |
| Impact momentum |
\(m_pv_0=(m_p+m_b)v'\) |
Momentum is conserved during the short inelastic collision |
| Swing energy |
\(\frac12(m_p+m_b)v'^2=(m_p+m_b)gh\) |
Mechanical energy is conserved during the ideal pendulum swing |
| Post-impact speed |
\(v'=\sqrt{2gh}\) |
Speed of block and projectile just after impact |
| Height from angle |
\(h=L(1-\cos\theta)\) |
Converts maximum swing angle into vertical rise |
| Projectile speed |
\(v_0=\frac{m_p+m_b}{m_p}\sqrt{2gh}\) |
Projectile speed before impact |
| Projectile mass |
\(m_p=\frac{m_bv'}{v_0-v'}\) |
Projectile mass when projectile speed is known |
| Impact energy loss |
\(K_{\mathrm{lost}}=\frac12m_pv_0^2-\frac12(m_p+m_b)v'^2\) |
Kinetic energy lost in the inelastic collision |
The ballistic pendulum uses momentum for the collision stage and energy for the swing stage. Do not conserve kinetic energy during the impact.
