Ballistic pendulum

Schematic representation

A ballistic pendulum is a mechanical device used to measure projectile velocities. It was invented by Benjamin Robins (1707–1751) in 1742 .

Working principle

Ballistic pendulum

A heavy block of wood (in the picture a wooden ball) is hung on a long wire or a rod so that it can rotate horizontally. The projectile to be measured is fired at the block of wood, gets stuck there and deflects it. By measuring the maximum pendulum deflection , the speed of the projectile fired can be calculated approximately .

formula

There are:

• ${\displaystyle M}$: Mass of the wooden block
• ${\displaystyle m}$: Mass of the bullet
• ${\displaystyle g}$: Gravitational acceleration
• ${\displaystyle l}$: Length of the pendulum from the suspension to the center of gravity of the wooden block

It is measured:

• ${\displaystyle \varphi _{m}}$: Angle of deflection of the pendulum to the rest position (vertical)

Then the bullet velocity is:

${\displaystyle v=\left({\frac {M}{m}}+1\right){\sqrt {2\,g\,l\,(1-\cos \varphi _{m})}}}$

This formula is a special case of the formula given below. However, it makes sense to give a complete derivation, since the angle is not always given. In practice it is much easier to measure the deflection. So you don't know the angle , only the horizontal deflection . ${\displaystyle \varphi }$ ${\displaystyle L}$

Derivation

It is an inelastic collision :

${\displaystyle (M+m)\cdot v'=m\cdot v}$,

where denotes the speed of the wooden ball after the impact. The speed is calculated with the help of the law of conservation of energy starting from the potential energy : ${\displaystyle v'}$

${\displaystyle E_{\mathrm {kin} }={\frac {1}{2}}(M+m)\,v'^{2}=(M+m)\,g\,h=E_{\mathrm {pot} }}$

For the amount of deflection we get: ${\displaystyle h}$

${\displaystyle h=l\cdot \left(1-\cos \varphi \right)}$

With the theorem

${\displaystyle \cos(\arcsin x)={\sqrt {1-x^{2}}}}$

we get the angle

${\displaystyle \sin \varphi ={\frac {L}{l}}}$,

which we use

${\displaystyle {\frac {1}{2}}(M+m)\,v'^{2}=(M+m)\,g\cdot l\,\left(1-{\sqrt {1-\left({\frac {L}{l}}\right)^{2}}}\right)}$.

Solving for finally yields: ${\displaystyle v'}$

${\displaystyle v'={\sqrt {2\,g\,l\cdot \left(1-{\sqrt {1-\left({\frac {L}{l}}\right)^{2}}}\right)}}}$

You put that in the formula of the inelastic collision. This gives the result:

${\displaystyle v={\frac {M+m}{m}}\cdot {\sqrt {2\,g\,l\cdot \left(1-{\sqrt {1-\left({\frac {L}{l}}\right)^{2}}}\right)}}}$

Individual evidence

1. ^ Encyclopædia Britannica Online

See also

• Twist (weapon)

literature

• Ekbert Hering, Rolf Martin, Martin Stohrer: Physics for engineers . 4th improved edition. VDI-Verlag, Düsseldorf 1992, ISBN 3-18-401227-1 .
• Beat Kneubuehl (eds.), Robin Coupland, Markus Rothschild , Michael Thali: Wundballistik. Basics and Applications . 3rd completely revised and expanded edition. Springer Medizin Verlag, Heidelberg, 2008, ISBN 978-3-540-79008-2 .