Ballistic pendulum

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A ballistic pendulum is a device for measuring a bullet's kinetic energy. The device is similar to a clock's pendulum: a mass at the end of a pivot arm, with the arm and mass free to rotate. A bullet is fired into the stationary pendulum, which captures the bullet and absorbs its energy.

The amount of energy delivered by the bullet can then be measured by how high the pendulum's mass is elevated as the pendulum rotates. However not all of the energy from the bullet is transformed to extra potential energy for the pendulum, some is used as heat and deformation energy. But the momentum of the system is conserved and we can use energy conservation after the bullet has hit the pendulum, which gives us (v denotes the speed of the bullet and u the speed of the system after the collision):

E_\textrm{kin} = E_\textrm{pot} \,
\begin{matrix}\frac{1}{2}\end{matrix} (m_\textrm{bullet}+m_\textrm{pendulum})\cdot u^2 = (m_\textrm{bullet}+m_\textrm{pendulum})\cdot g\cdot h
u = \sqrt{2\cdot g\cdot h}

We can then use momentum conservation to get the speed of the bullet, as:

p_\textrm{before} = p_\textrm{after} \,
m_\textrm{bullet}\cdot v = (m_\textrm{bullet}+m_\textrm{pendulum}) \cdot \sqrt{2\cdot g\cdot h}
v = \frac{(m_\textrm{bullet}+m_\textrm{pendulum}) \cdot \sqrt{2\cdot g\cdot h}}{m_\textrm{bullet}}

The pendulum usually strokes a marker that records the maximum height of the pendulum during the test.

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