Inertia and momentum

Question

Question #1: Why does speed have nothing to do with inertia?

Question #2: If a car hits a steel wall and stops, where did the momentum go?

Answer 1

Momentum is $p = m v$, so it does have to do something with speed in the sense that it is proportional to it. However, you can have a different momentum with the same speed and vice versa.

Take a bullet that has a mass of 3 g and a velocity of 370 m/s. Then its momentum would be about 1 kg m/s. You get the same momentum if you take a pack of flour (1 kg) and let it have a velocity of 1 m/s. Both can transfer the same momentum to something else. If you use both to knock something off, the force would be comparable.

If a car hits a steel wall and stops, its momentum goes into the earth, which will rotate a tiny bit faster. The total momentum is conserved, so the wall (which is attached to the earth) has to carry the momentum that the car had previously. Since the mass of the earth is enormously greater than the car, the difference in speed is almost zero.

Answer 2

Why does speed have nothing to do with inertia?

It depends on what you mean by the word "inertia". When used colloquially, "inertia" sometimes means "mass", but at other times, "momentum" (mass times velocity). So depending on meaning, inertia has nothing to do with velocity or it is intimately coupled with velocity. I suspect you think of "inertia" as momentum, but have read or heard somewhere that it means something different, i.e., mass. Physicists tend to use the words mass and momentum to avoid confusion.

Newton's first law is sometimes called the law of inertia. In this sense, inertia is neither mass nor momentum. It is simply the tendency of an object to follow a straight line trajectory at a constant speed, unless acted upon by an external force.

There's a potential problem here. If you yourself are accelerating or rotating, you will not see a force-free object follow a straight line trajectory at a constant speed. The modern view of Newton's law of inertia is that it introduces a preferred set of reference frames, the inertial frames of reference. A reference frame is inertial if Newton's law of inertia holds for all force-free objects.

If a car hits a steel wall and stops, where did the momentum go?

This is a good question!

It went into the wall, and then into the Earth. Imagine a pure inelastic collision between a spaceship of mass $m$ going at velocity $v$ and a big wall in space of mass $M$ that is motionless with respect to the observer of the collision. After the collision, the observer will see that the spaceship+wall system has a velocity $V=\frac m{M+m} v$. When $M$ is many, many orders of magnitude larger than $m$, the final velocity will be very small. When $M$ is many, many, many orders of magnitude larger than $m$, the final velocity will be so close to zero so as to be unobservable.

Getting back to the problem at hand: The mass of the Earth is over $10^{21}$ times that of a typical car. That falls in the class of "many, many, many orders of magnitude".

Answer 3

If a car hits a wall momentum would be lost in form of sound and heat.