Again, momentum and kinetic energy.?

Question

Say a car crashes into a mountain at 3mph.

If it were going 20 times faster then that it would produce a collision *400* times worse, with 20 squared times the energy.

Why is the car still only 20 times harder to stop?

Answer 1

It is not correct to say that the car is still only 20 times harder to stop. This is because the force required to stop the car is the rate of change of momentum.
Lets take an example-
You are travelling at 3mph and suddenly see a barrier 2 meters away. You apply the brakes and stop just in time.
Now if you are travelling at 60 mph and you want to stop within 2 meters you would have to apply the force which is 400 times the original force and not 20 times. This can be seen from the basic formula
v^2 - u^2 = 2as
where s is the distance.

Answer 2

Actually, if we used rocket engines to slow down the car, it will take 400 times as much fuel to do the job, not 20. The energy required to boost the kinetic energy of an object goes up with the square of the velocity, or proportional to the kinetic energy, and the same is true in reverse.

Answer 3

F=MA
The force required to stop an object in a particular unit of time is indeed proportional to the speed of the object.
The energy used is the force times the distance moved.
D= 1/2A*T*T
so E = M*A*A*T*T=M*A*T*A*T=M*V*V
The energy is therefore proportional to the velocity squared.
In reality the time to stop a car running into a mountain at 3 Miles per hour is really less than the time it takes to stop a car traveling at 60 miles per hour. The deforming of the mountain and car would be significantly greater for a faster car.