Ouch.
I just read this thread and it gave me a headache…and I studied physics.
Let me see if I can boil things down.
The key concepts are:
- Conservation of Momentum
- Elastic vs. Inelastic Collisions
- The Principle of Relativity
- Cars vs. Walls (ambiguity in the OP)
Taking the side of the underdog, if we interpret the OP’s statement:
*I have a knucklehead friend I am unable to convince of a fairly simple physics problem. He contends if two cars of equal mass (discounting crumple zones and the like) both traveling 35 mph meet head-on, the resulting collision is multiplied to a 70 mph crash. *
in a manner charitable to the knucklehead, we can presume he (the knucklehead) meant that for two cars in a head on collision, it is the same for each as though they had been stationary and hit by another car moving 70mph (or as if they had been moving 70mph and hit a stationary car).
I presume this is what the knucklehead meant, otherwise, the progression would go as follows:
Two cars colliding at 35mph multiply to a 70mph collision.
What is a 70mph collision like?
Well: two cars colliding at 70mph multiply to a 140mph collision.
…
…
Two cars colliding at 5600mph multiply to a 11200mph collision…
…and so on…
You can start at 1mph and end up at C (the speed of light) or beyond.
Its a rare knucklehead who would suggest all collisions are equivalent to impact with infinite velocity.
Using the Principle of Relativity (also useful for deriving the Theory of Relativity, but let’s save that for another day), it becomes clear that the knucklehead is more head than knuckle. If you have Car A and Car B, one moving East at 35mph relative to you and one moving West at 35mph relative to you; without an external reference point, it is indistinguishable to Car A whether:
- he is moving at 70mph and Car B is stationary;
- he is stationary and car B is approaching at 70mph;
- they are each moving towards each other at 35mph;
- he is moving at velocity X and Car B is moving at velocity Y, where the difference between X and Y (how fast they approach each other) is 70mph.
If you’ve got a problem with that, take it up with Einstein (actually, since we’re talking principle of relativity, rather than Theory of Relativity, take it up with Mach, but that’s not important right now).
(All of this also applies to Car B.)
Now, the trick is to remember to stay in the same reference frame after the collision.
If you consider case:
- both cars appear to be moving at 35mph in the direction of Car A’s original motion after the collision;
- both cars appear to be moving at 35mph in the direction of Car B’s original motion after the collision;
- both cars appear stationary after the collision;
- left as an exercise to the reader. (That’s physics lingo for: I’m too lazy to write out the equations.)
Conservation of momentum (the sum of both velocities (speed + direction) has to sum to the same number before and after…) is the only thing we need to worry about here, because we are dealing with an inelastic collision. In an elastic collision (billiard balls) things bounce off each other and both momentum and energy are conserved. In an inelastic collision (car crashes, football tackles, billiard balls made of play-dough) objects fuse and only momentum is conserved (note that in each of the four reference frames, there is the same total momentum before and after…but you cannot compare momentum across different inertial reference frames).
It was actually Bigtrout who brought up the brick wall.
His friend, if we stuck to talking about cars, would be correct, provided the second car in the 70mph collision wasn’t some magical car that began stationary, and remained stationary after the collision without being acted upon by an outside force.
By changing the problem from one of cars to one of a car and a wall, Bigtrout basically did just that: the wall is a “magic car”.
If you collide with a wall at 35mph you feel the same force as if you hit a (neglecting crumple-zones and the like as we have been in previous posts) car which is held perfectly still both before and after the collision. The problem is, the wall needs to bring outside forces to bear to stop the moving car (being fixed to the ground, a mountain, whatever); the same thing would happen if you had a car that was stationary both before and after but in the latter case, it is more obvious that an outside force has to hold the car still.
Note that there is no reference frame where car B is stationary both before and after the crash!!!
That’s why we have trouble comparing the two.
Now, since the wall brings you to a complete stop, neglecting differing physical properties of walls and cars, you as a driver in car A moving 35mph experience the same phenomenon if you:
- Drive into a brick wall at 35mph and come to a complete instantaneous stop; (techically no such thing, but just assume the same stopping time for all the collisons)
- Drive into an oncoming car at 35mph, bringing you both to a complete instantaneous stop;
- Drive into a stationary car at 70mph, and both you and resultant wreckage (both cars) continue on at 35mph;
- Drive into a stationary “magic car” at 35mph which does not move when you hit it, bringing you to a complete stop.
So two cars driving into each other at 35mph is the same as hitting an initially stationary car at 70mph (what your friend probably meant) but not hitting a wall at 70mph (the words you put in his mouth).
I think when you put it that way, Bigtrout, your knucklehead friend doesn’t sound like so much of a knucklehead.
Tell him he owes me a beer.
I say you owe him a new moniker.