# Should force impact moving objects less than ones at rest?

## Main Question or Discussion Point

I'm sure I'm not thinking of this the right way. I'm hoping someone can see the error in my logic.

Should a force have less of an impact on a moving object than one which is at rest?

An object which is in motion has a momentum given by p=mv. A force F acting (say in the opposite direction) on this object need to overcome the object's momentum.

The same force acting on an object at rest needs not overcome any momentum (thought I guess the object here has inertia).

So wouldnt it make sense for the force to have more of an effect on the object which was at rest than one which was in motion? (By 'effect' I suppose I mean difference between initial and final velocities vf-vi)

If this is not the case, then does it mean inertia and momentum are the same thing?

## Answers and Replies

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A meaningful "effect" would be a change in the object's momentum when considering forces. It doesn't matter whether the object is moving or not; the (resultant) force will change its momentum. If the object is at rest it will start to move and gain momentum, if it is already moving, it might move faster and gain momentum, or be slowed down and lose some. (It might even just change direction)
Either way, what the force has done is to change the object's momentum. (By an amount = force times time)

At non-relativistic speeds, the acceleration of a body under the influence of force F remains the same irrespective of its speed. a= F/m. No speed component here.
If this is not the case, then does it mean inertia and momentum are the same thing?
Inertia is the nature of any particle- To remain at rest or uniform motion if no force acts on it. Momentum is a quantity. You can attribute a value to it. Because of inertia (First law), momentum "happens" to remain constant if there's no external force.

Thanks for your replies. I have some lingering issues...

A meaningful "effect" would be a change in the object's momentum...
But a change in an objects momentum is equivalent to a change in its velocity since
$$p=mv$$

Either way, what the force has done is to change the object's momentum. (By an amount = force times time)
What happens if the force is applied for a very short period of time, say its given by
$$F(t)=F\delta(t)$$

where $$\delta(t)$$ is the Kronecker delta function.

How does this change the momentum?

v = a*t + vo

If the force is applied for only a short time, then the momentum change is based on that above. .

Actually momentum can be thought of the integral of force in change in time, although the term impulse is used instead of momentum.

If you read the wikipedia article on impulse it might help you out.
"A small force applied for a long time can produce the same momentum change as a large force applied briefly."

Your original question was about the "impact" of a force on an object, and if it is less when the object is moving.
The answer is; it depends what you mean by "impact". The result of a (net) force acting on an object is a change in its momentum. The change is determined by that force times the time it acts. Ft. The change in momentum is Ft. Yes, this usually involves just a change in the object's velocity, but it is the change in momentum whereby we measure the "impact" of the force. (And yes, this equation assumes a constant force. You need the integral of force over time in the more general case. But the argument is the same.)
There is no difference between a moving object or one at rest from the point of view of the change in momentum. If the object is moving the force could cause it to stop moving, if the object is at rest the force could cause it to start moving. In both cases there is a change in momentum (and velocity) and in both cases this (the change in momentum) is given by force x time.
The inertia you speak of applies equally to the tendency of an object at rest to want to stay at rest, as it does to an object in motion wanting to stay in uniform motion in a straight line. The force has to overcome inertia in both cases.