# Modified Car and ditch paradox

1. Aug 30, 2009

### I_am_learning

Suppose there is a car moving on a road in relativistic speed. The road has a ditch (hole) which is litter bigger than the car when both are at rest.

Modification: Assume that the car will fall on the ditch if all of its tyres are in the air at the same time. If two of the rear tyres rest on the road and two of the front tyre hang on the air, the car won't fall in the ditch; it can move as if all of its tyres are on road.

Now the problem is, car seem to fall on the ditch on the
roads perspective whereas in the cars' perspective it can pass the ditch without trouble.
The question is Will the car fall in the ditch?

(Though I have much understood much of S.R. I got stuck on this. I know that: though in the road's frame of reference the two rear and the two front wheels seem to be in air at the same instant its not the case in the car's frame of reference. In the car's perspective they occuur one after the other.
But anyway, it can't be that --> for the car it can escape the ditch AND for the road it can trap the car.? )
Have attached picture for visualization.

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2. Aug 30, 2009

### Staff: Mentor

Re: Modified Car and ditch paradox!!!

"at the same time" according to whom? Since simultaneity is frame-dependent, your modification is unphysical.

3. Aug 30, 2009

### MikeLizzi

Re: Modified Car and ditch paradox!!!

This is a variation of a familiar problem. And it suffers from the same weakness. Your modification to the problem contradicts reality. That makes any solution meaningles. The truth is, this is a very difficult Mechanical Engineering problem. The other side of the ditch must be lower by a certain amount for the car to make it accross no matter whose Physics you use (Newton's or Einstein's). That means it will always crash into the other edge. So why not just mark a spot on the opposite edge where it hits. Then the problem is to verify that it will hit the same spot wheather you observe the action from the ditch or as a passenger in the car.

Hint: Remember that velocities don't add the same intuitive way using Special Relativity. The vector sum can never reach "c". So, the faster the car goes along the road, the less downward velocity is available for it to acquire when it's hovering above the ditch. At .9c it will drop much less than expected before it hits the opposite edge. Now switch and be the observer in the car. The car is at rest and can acquire all the downward velocity it wants once its above the ditch. But the ditch is moving at .9c and is much narrower. So the car hits the opposite edge in the same spot. Make sense?

4. Aug 30, 2009

### I_am_learning

Re: Modified Car and ditch paradox!!!

To tell According to roads view.
To make the modification sensible assume that on the both side of the roads are pressure sensors. As long as at least one sensor is kept pressed all is fine. But whenever there is no pressure in all of them at the same instant (from the road's frame) a robotic arm that emerge from the ditch will snatch the car.

I think this is the answer now ? --> The car's driver, Having known the arrangement of the road, but now happy about the contracted ditch's length, would try to cross the ditch casually. But alas, the contracted length is only a mirage, he is snatched away!!!