Lead bullet vs a Rubber bullet?

Our teacher says that rubber bullet is more effective than a lead bullet for knocking down a bear/human/whatever. He says that change in momentum for rubber bullet is greater than that for lead bullet. So more force is exerted by rubber bullet. But I don't get him and I don't think he is right too. Can you get this straight for me?

• Think about the difference in the two collisions. Which one transfers more momentum to the human? Apr 15 '15 at 16:52
• I am trying to get what you are saying, point me out if I am wrong, in ideal case, lead bullet transfers large amount of momentum to the target but in case pf rubber one, the rubber bullet just bounces back with same velocity, i.e. no momentum is tranferred?
Apr 15 '15 at 17:05
• Something rebounding transfers, by conservation of momentum, more momentum than something that is simply stopped. Apr 15 '15 at 17:06
• HOw??????It's a bit confusing
Apr 15 '15 at 17:14
• Ok I got it anyway
Apr 15 '15 at 17:15

First, momentum $p$ is conserved in the system during a collision:

$$\sum p_{before} =\sum p_{after}$$

So, the lost momentum of the bullet must be transfered to the body (since there is nothing else to absorb it).

Second, force is the change of momentum over time (Newton's 2nd law):

$$F=\frac{\Delta p}{\Delta t}$$

So, the larger the moment transfer (the larger the change in momentum), the bigger the force.

And remember how momentum is calculated:

$$p=mv$$

Now, let's consider the two bullets:

• Lead: It penetrates, so it looses some speed and continues with less speed. It goes from a start speed $v_1$ to a smaller speed after impact, $v_2$. Momentum change is $$\Delta p=p_2-p_1=mv_2-mv_1=m(v_2-v_1)$$

• Rubber: It bounces back, so it looses all it's forward speed and gains backwards speed to move backwards. The start speed is still $v_1$ but the final speed is negative (in the opposite direction), $-v_2$. Momentum change is $$\Delta p=p_2-p_1=m(-v_2)-mv_1=m(-v_2-v_1)=-m(v_2+v_1)$$

Now which momentum change $\Delta p$ was bigger (disregard the sign, because we don't care about the direction)? The rubber bullet's! It had it's speed changed from positive to negative, which (since they had same start speed) must be more than a speed change from positive to just less positive.

You might object, though, and claim that the rubber bullet must exert less force since it doesn't penetrate, while the lead bullet does.

But remember then, that penetration is not about force, but pressure.

• If I slap you with my hand, you might get a blue mark, but if I slap you with a nail in my hand with the same force, the nail will go through and penetrate.

In the same way, the rubber bullets might indeed hit harder, but they are usually rounded and deforms elastically at impact so that the contact area is much larger. The pressure, which is force per area, is then still smaller and tolerable of the skin.

The lead bullet with the sharp pointing nose, hits with its whole force at this tiny area at the nose tip, like a needle picking straight through.

Your target may not stop a very dense, compact bullet: It may exit, departing with some (possibly large) fraction of its initial momentum. If your bullet does not penetrate a target as well, as a less dense rubber bullet might (depening on circumstances including its speed and properties of the target), it might be stopped completely (either stuck in the target or dropping down in front of it), or even recoil after impact. Then its entire momentum (and in case of a recoil even more than that) will be transferred onto the target. This may be what your teacher is trying to get at.

Note that this argument may not be very relevant in practice: The penetrating bullet will injure or kill with greater likelihood and in that sense impart more stopping power. It is also likely to be fired with more kinetic energy, have a higher mass, and possibly carry a lot more momentum than a rubber bullet likely designed to be "less lethal," the primary reason to use these otherwise less effective (less damaging) bullets in the first place.