Newton's 2nd Law: Zero Acceleration Problem

1. Nov 30, 2009

leibniz33

Hi all, here is a problem that has popped into my head every so often for the last year or so:

A small, low-mass body is moving at a constant velocity of 1000m/s through a vaccuum, so it has no acceleration or deceleration i.e. a=0. Now let's say you are an astronaut and you are floating directly in the path of this oncoming body. You are completely stationary i.e. you have zero velocity. The small body collides with you. According to Newton's 2nd law, Force = Mass x Acceleration, so if you want to calculate the force F carried by the small body that hits you, you should be able to use this formula. However, if a=0, then F = m(0), therefore F=0. How can a body moving at 1000m/s hit you and impact no force on you?

Bear in mind that this is intended purely as a hypothetical. Of course in actual space, the gravity of nearby large bodies like planets/moons/stars would give a certain acceleration to the small body. However, even if a very small moon was nearby and it gave the small body an acceleration of 0.01m/s^2, this would still result in a very small F value, much less than would be expected of something travelling at 1000m/s. I'm sure there is a very simple explanation for this, but i haven't been able to find one yet.

Cheers.

2. Nov 30, 2009

Staff: Mentor

When it hits you the acceleration will no longer be 0.

3. Nov 30, 2009

elvinc

The low mass body is not accelerating because no overall force is acting on it.

However, the body has momentum.

When bodies collide, elastically or inelastically, the total linear momentum of the system is conserved (assuming not rotation/spinning) . Some of the momentum of the moving body will be "transferred" to the stationary body. How much momentum is transferred depends on the coefficient of restitution between the two bodies.

As far as the force is concerned. Yes the hit body accelerates so a force has acted on it - but it is not a constant force. The relevant equation is (I believe)

$$\int F dt$$ (impulse) = change in momentum of either body in 2-body system
http://en.wikipedia.org/wiki/Impulse

Clive

4. Nov 30, 2009

Fightfish

Yup, in addition to the comments already posted, the key thing to resolving this 'paradox' is that when the moving object collides with something else, it decelerates.

5. Nov 30, 2009

leibniz33

Thanks very much, that makes clear sense now that I think about it!

So does that mean that the small body in the problem technically carries no force until it receives an acceleration by colliding with another body or being affected by an external force? That's an unusual concept..

6. Nov 30, 2009

rcgldr

Both the small and large body carry momentum, not force. A net force only results when the momentum is changed. By momentum, I mean mass times velocity, so this includes circular movement, where speed is constant, but the inwards acceleration results in constantly changing velocity.

7. Nov 30, 2009

elvinc

That's the nub of your misunderstanding methinks

In Newtonian mechanics (and probably other mechanics that use the idea of force) bodies do not carry force, forces act on bodies (if they are external) and cause a change in their motion / momentum either acceleration or deceleration. Newton 1 - once a body has been accelerated by an external force, and that external force has been removed, the body no longer needs a force to keep it moving with constant velocity in a straight line.

Clive