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Classical Mechanics Level 3

Which bullet will result in greater damage for a bear, a lead bullet, an iron bullet, or a rubber one?

Details and Assumptions:

Both bullets are fired from the same gun, and experience the same force for the same amount of time.
We take the energy of the bullet as a proxy for how much damage it does to the bear. E.g. assume the bear wears a bullet proof vest, so that the damage is done by dissipating the bullets energy.

rubber bullet both will have the same impact lead bullet iron bullet

1 solution

Bogdan Profir
Jan 12, 2017

Because of the densities of the three materials, we can write that $m_L>m_i>m_r$ , where L,i and r represent lead, iron and rubber. Obviously, this relationship holds because we assume that all three bullets have the same shape (have the same volume) and hence they only differ through their densities which will cause the above relationship to be true.

For simplicity we could also make the assumption that the force exerted by the gun will be constant throughout the time it will be acting on the bullet (generalizing and using integrals should give exactly the same result). Therefore, we can write:

$v=(\frac{F\Delta t}{m}$ ) - which results from the conservation of momentum

The kinetic energy of any bullet can be written as:

$E=(\frac{mv^2}{2}$ )

By substituting the expression for velocity, we can see that the kinetic energy becomes:

$E=(\frac{(F\Delta t)^2}{2m}$ )

Because the mass exists only in the denominator in the equation above ( $F$ and $\Delta t$ are the same for all bullets), it can be inferred that the lower the mass, the greater the resulting kinetic energy will be. Consequently, the rubber bullet (which weighs the least based on its density) will have the highest energy and therefore the highest impact of the three.

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