lobuxracer

iTrader: (2)

Sorry, but the wall does move. It has to. When you jump from the surface of the planet, you both experience an equal force pushing away from each other, but the mass of the planet is so great the amount of movement is infinitesimal. Same with the wall. It surely moves, just not noticeably because of the mass behind it.

It would be interesting to see the results of a 50/50 and a 100/0 head on crash for a lot of reasons...I think the wall changes things a lot compared to two cars.

Infra

iTrader: (1)

*edit* Sorry, read that wrong. Yes for a stationary car, no for a wall.

The wall moves, but as it's anchored to the earth, the earth must move for the wall to move. Since the mass of the earth is so much larger than the mass of the car, we can assume that the distance the wall moves is negligible. Of course, deformation of the wall is difficult to account for, but again, the car deforms so much more than the steel wall that the deformation of the wall is negligible - assuming the wall does not move or deform at all would give an answer that is pretty close to reality.

This is a nearly inelastic collision - the car and wall stick together (not really but it's pretty close as opposed to the car bouncing off and travelling in the opposite direction at the same initial velocity). Conservation of momentum, m1v1 + m2v2 = m3v3, where m1 is the mass of the car, m2 is the mass of wall+earth, and m3 is the mass of car+wall+earth, shows us that when the car sticks to the wall, which is also stuck to the earth, velocity must go to near zero (which is common sense when anything runs into a wall).

The issue here is what is absorbing the energy of the impact. In the case of 2 cars, we can assume each car absorbs 1/2 of the energy. If you were to drive a car of the same mass at twice the velocity into a stationary car of the same mass, the result would be the same, only the now double-car would have a non-zero final velocity. In an ideal inelastic collision, it would have a velocity of 1/2 of the initial, or in this case 50 mph.

When the car hits a wall, the wall (and the earth it is attached to) do not absorb any of the energy of the impact, meaning that the car now must absorb twice the energy.

Just like how an airbag is essentially a wall for your chest and face but absorbs the majority of the energy of the impact, it matters how much energy the steel wall absorbs through deformation.

For a more thorough explanation with all the math, see this.

PhantomZX

I saw your post, but as I stated, speed is relative. When you say one car is at rest and the other is going 100mph, that's in relation to the surface of the earth. When two cars are heading toward each other at 50mph, one car is actually traveling at 100mph relative to the other. What does the relation to the surface of the earth have anything to do with the impact between these two cars traveling at 100mph in relation to each other?

Infra

iTrader: (1)

I misread your post. Please see my correction.

PhantomZX

Got it, but what about the conundrum I brought up about the relative speeds.

We know that e=mv^2 so let's say a car at rest on the surface of the earth has no velocity, so it's force=0. Let's just plug in some nice round numbers in to the equation and say a car with a velocity of 100 and a mass of 100 has energy value of 1,000,000.

Now lets say relative to the surface of the earth, each car has a velocity of 50 so their energy value is 250,000. Thus, both cars have a total energy value of 500,000.

Where did that missing 500,000 units of energy go?

PhantomZX

After thinking about it a bit more, I think I do know where it goes. The missing 500,000 units of energy is relative to the speed of the surface of the earth and the movement of the two masses after the impact should contain the 500,000 units of energy. When one car hits the other, They will both move (the one at rest will now be moving, and the one originally at speed 100 will be going at a slower speed but still be moving).

Infra

iTrader: (1)

Well, first, the equation is KE = 1/2 mv^2, but that doesn't really matter.

You are assuming that in the collision, the end velocity is zero. Furthermore, to calculate the end velocity, you must use conservation of momentum, where momentum is equal to m*v. The end velocity is not zero because the 2nd car is not stuck to the earth. The end velocity would be, in an ideal inelastic collision, 1/2 of the initial velocity.

As for your question, that 500,000 units of energy is the kinetic energy of the combined cars moving at the final velocity.

Allow me to go out on a limb here - that still probably doesn't answer the question for you. "How does each situation have a different amount of initial kinetic energy when speed is relative?" is what I think you are truly asking. I believe you are confusing your frames of reference.

Our frame of reference is the earth, assumed to be non moving. A car traveling twice as fast has 4 times the energy (v^2), therefore 2 cars at half the speed have half the energy. When you say motion is relative, certainly from the frame of reference of one car, the other is traveling towards it at 100 mph in both cases, but the frame of reference of us, the observer, is not the same as the frame of reference of the car.

If our frame of reference is the earth, the cars, both moving at 50 mph, collide and Vf (final velocity) is zero.

One car moving at 100 mph, Vf is 50 mph.

If our frame of reference is now moving at 50 mph identical to the first car... the two cars, each moving at 50 mph, *appears* to us to as the first car not moving, and the second car moving at 100 mph. They collide and now the combined mass' Vf is 50 mph.

One car moving at 100 mph, *appears* to us as each car travelling at 50 mph, collide, and Vf is zero.

The initial value for kinetic energy is 100% dependent on your frame of reference. What matters is the amount of energy dissipated in the collision, and this remains consistent in each case.

Infra

iTrader: (1)

We could be travelling at 1,000 mph, which in the case of each car moving at 50 mph, would make the first car appear to be moving at 1,050 mph, and the 2nd at 950 mph in the same direction as the first car, and the kinetic energy of each would be ridiculously high, yet the energy dissipated in the collision remains the same.

RXSF

with the yellow and red car at 50 and 100 mph, shouldnt they have rammed them against a collapsable object instead of a steel wall?

PhantomZX

The myth they were trying to test was Jamie's original statement that two cars impacting each other head on at 50mph each was equivalent to one car impacting a wall at 100mph.

syzygy

This is a great link, thanks for the resource . It's all coming back now, masses colliding into each other and sticking together (i.e. "perfeclty inelastic collisions," if I recall correctly), momentum being conserved regardless of the type of collision. Very nostalgic indeed.

I feel like picking up my Feynmann lectures on mechanics and just brushing up on all of this again after so many years.