use conservation of momentum to discuss automobile safety, including the roles of seatbelts and whiplash?

Question

from rear-end collisions.

Answer 1

Don't figure that out. Christ if someone solves the whiplash problem a lot of attornies are going to starve.

Answer 2

Mv + mV = (M + m)u for a perfectly inelastic collision of masses M and m with initial velocities v and V. u is the post-impact velocity of the stuck together masses (M + m). If the collision is perfectly elastic, Mv + mV = Mv' + mV' where v' and V' are after impact velocities that depend on the angle of impact (e.g., head on, glancing). In reality, the collision will be somewhere between perfectly inelastic and perfectly elastic.

Now suppose you (m) and your car (M) are moving along at u velocity. Your momentum would be L = (M + m)u. But after you doze off and that lamp post runs into you, the mass of your car comes suddenly to a halt.

If you have your seat belt tightened; so do you come to a sudden halt. In which case, (M + m)u = (M + m)v after the impact; so that we see that u = v. That is, your velocity and that of the car right after impact will be no faster than the speed at which you crashed into the lamp post.

The after impact kinetic energy for this scenario would be ke = 1/2 (M + m)v^2. The lamp post would absorb that kinetic energy and crumple, as would a lot of your car. The expectation is that the engine block, hood, and front of the car, along with the lamp post, will absorb sufficient ke that the crumpling will cease before it reaches you safely locked into your driver's seat.

If you have not fastened your seat belt (tsk, tsk), the car comes to a halt, but you continue on (into the windsheild and beyond). In which case, (M + m)v = mV; where V is you flying into the windshield. Your kinetic energy as you smash into the glass would be KE = 1/2 mV^2. ouch.

Note that V = v (M + m)/m. Thus, for example, if your car weighs 1,800 pounds and you weigh 200 pounds, you velocity flying out of your drivers seat will be V = v (M + m)/m = v (2000/200) = 10v or ten times the velocity at which you drove into the street post.

To see what this means to your surviving the crash consider the two kinetic energies (KE and ke). KE = 1/2 mV^2 and ke = 1/2 (M + m)v^2; so that KE/ke = 1/2 mV^2/(1/2 (M + m) v^2) = 1/10 (100 v^2/v^2) = 10 ke. Thus, you alone have ten times the kinetic energy of you and the car together. And the only way that gets dissipated is by crashing through the windshield and breaking up your body and vital organs.

This is an amazing result. If you fasten your seat belt, in this example, there would be 1/10 the kinetic energy to expend when compared to the KE of you alone flying through the windshield because you did not strap in. I'd say using a seat belt makes a lot of sense.

As to whiplash, that results from Newton's first law, not the conservation of momentum. In this case, a mass (m), which is you is traveling at a steady velocity (v) while sitting in the driver's seat. All of a sudden, the car gets rear ended.

That results in a force on your body (through the seat), but not on your head. Thus, due to Newt's law, your head continues on with that velocity (v) while your body is accelerated (by the rear ender) to v' = v + at = v + (F//m)t. So your body lurches ahead at v' while your head stays at v < v'. As a result, the distance your head travels = d = vt < v't = d' is less than the distance the car travels in time t. The effect is that your head moves backward relative to the car...and that can give you whiplash, which is simply a neck injury because of the sudden jerking backward.