classical mechanics - Car accident, put in park or neutral?

I was waiting on a red light the other day and was wondering.

If I'm in my car, not moving and I see a car that's going to hit me from behind. Would I (my body) be safer if I put on the break or if I put the car in neutral?

I assume there's no car in front of me and there's no posibility to be hit by an other car. Also, the car crashing into me is going at a good speed.

Answer

I agree with RedGrittyBrick's answer i.e. that you should apply the brakes, but it's interesting to see why.

The easy situation to analyse is when you don't apply the brakes so (ignoring rolling friction and air resistance) you have a collision between free bodies. Suppose the other car hits you at a velocity of $v$, then the force you feel is the same as if you reversed into a solid object at a speed of $v/2$. To see why this is consider the centre of mass frame. If the cars are the same mass, in the COM frame both cars are moving towards each other at a speed of $v/2$ (so the relative speed is $v$) and assuming a symmetrical collision the cars come to a stop at the centre of mass. hence it's like a collision with an immovable object at $v/2$.

Actually there may be circumstances when this type of collision is best and you should leave the brakes off. The force in a collision is proportional to $v^2$, so reducing the collision velocity by a factor of 2 reduces the force by a factor of 4. There will be some velocity $v_{min}$ below which the collision won't do you any harm, so as long as $v_{collision} < 2 \space v_{min}$ leaving the brakes off won't do you any harm and it will lessen the damage to the cars. I am assuming here that leaving the brakes off doesn't result in you being propelled through the red light into the path of an 18 wheeler :-)

The situation is more complicated when the brakes are on because the collision takes place in three stages:

1. 
   

   the other car hits you and starts compressing your car's crumple zone.

   
   
   


2. 
   

   the collision force exceeds your braking force and your car starts accelerating. Your car's crumple zone is still being compressed.

   
   


3. 
   

   the crumple zone is fully compressed so both cars are now moving at the same speed and your brakes are slowing both cars.

   
   

During phase 1 your car is stationary so you feel no force. If the collision is low speed the other car may come to rest before your car starts to skid, and you feel no force at all.

However all but the most trivial collisions are likely to apply more force than your brakes can resist, and you enter phase 2. To calculate the force you feel in phase 2 is quite involved as you'd have to know the force distance curve for compression of the crumple zone. I did Google to see if I could find this data, but without success. Anyhow, it should be obvious that the force during phase two will be less than you'd feel if the brakes weren't on.

Phase three is interesting because it's where you get the whiplash. Assuming your headrest is properly adjusted your head won't move much in phase two. However when you enter phase three your own brakes will jerk you forward. To reduce this (and as above assuming there are no 18 wheelers around) you should release the brakes.

So for low speed collisions you should leave the brakes off, but for high speed collisions hit the brakes during the collision and release them as soon as the two cars have stopped moving relative to each other.

Having said all this, the collision you describe happened to me a few years ago (I'm sure it's hapened to lots of us) and I hit the brakes and held them on. I'm happy to report I suffered no harm, though my car was a write-off. I would advise my children to always hit the brakes and leave them on. Risking whiplash is better than being pushed across the junction or into the back of the car in front of you.