The elusive difference between force and impulse

It's hard to think of a physical system involving a force that acted for zero time. However I think it's useful to consider a collision, perhaps between two billiard balls.

When the balls collide they change momentum. We know that the change of momentum is just the impulse, and we know that the impulse is given by:

$$ J = \int F(t)\,dt $$

where I've used an integral because the force is generally not be constant during the collision.

If we use soft squidgy balls then the collision will take a relatively long time as the balls touch, then compress each other, then separate again. If we use extremely hard balls the collision will take a much shorter time because the balls don't deform as much. With the soft balls we get a low force for a long time, with the hard balls we get a high force for a short time, but in both cases (assuming the collision is elastic) the impulse (and change of momentum) is the same.

When we (i.e. undergraduates) are calculating how the balls recoil we generally simplify the system and assume that the collision takes zero time. In this case we get the unphysical situation where the force is infinite but acts for zero time, but we don't care because we recognise it as the limiting case of increasing force for decreasing duraction and we know the impulse remains constant as we take this limit.

I'm not sure it's helpful to think about the gravitational force, because I can't see a similar physical system where we can imagine the gravitational force deliverting a non-zero impulse in zero time.

Response to edit:

In you edit you added:

If I got it right, you are saying that we must consider it impulse when t=0?, else it is force.

I am saying that if we use an idealised model where we take the limit of zero collision time the impulse remains a well defined quantity when the force does not.

However I must emphasise that this is an ideal never achieved in the real world. In the real collisions the force and impulse both remain well behaved functions of time and we can do our calculations using the force or using the impulse. We normally choose whichever is most convenient.

I think Mister Mystère offers another good example. If you're flying a spacecraft you might want to fire your rocket motor on a low setting for a long time or at maximum for a short time. In either case what you're normally trying to do is change your momentum, i.e. impulse, by a preset amount and it doesn't matter much how you fire your rockets as long as the impulse reaches the required value.

Response to response to edit:

I'm not sure I fully grasp what you mean regarding the book, but the force of gravity acting on the book does indeed produce an impulse. Suppose we drop the book and it falls for a time $t$. The force on the book is $mg$ so the impulse is:

$$ J = mgt $$

To see that this really is equal to the change in momentum we use the SUVAT equation:

$$ v = u + at $$

In this case we drop the book from rest so $u = 0$, and the acceleration $a$ is just the gravitational acceleration $g$, so after a time $t$ the velocity is:

$$ v = gt $$

Since the initial momentum was zero the change in momentum is $mv$ or:

$$ \Delta p = mgt $$

Which is exactly what we got when we calculated the impulse so $J = \Delta p$ as we expect.

A force is not an impulse, it's a force. A force can exist without producing any work, stresses in materials are typically generated by forces applied to the same solid that oppose themselves and therefore do not produce any work. Some forces require contact, some forces don't (infinite range, decreasing with distance). As long as you're under the influence of a force and it is not counteracted, you will get work from it. If the duration of that influence is short with respect to your timescale, the linear momentum gained from it can be considered an impulse.

I personally see an impulse as a "discontinuity" of linear momentum for the timescale considered. If you could turn ON and OFF gravity for short durations, and you were dealing with small forces, you could end up with an impulse of gravity. If there is such thing as a strict definition of impulse, I would reckon the book would have an initial impulse if it was fired from a cannon; gravity keeps working over time (and is quite weak to create significant brutal changes in momentum). A collision is intrisically short and could be considered an impulse as well (in a different direction).

Example that I find relevant: 2 models exist for orbital manoeuvres: impulse burn where the time is actually an infinitesimal duration, and finite burn where by the time the manoeuvre is finished the spacecraft has already moved significantly. Either way what we would call the "impulse" is a change in velocity, which is a shorthand for linear momentum when the mass is known. In that particular case impulses are ideals that make calculations easier, for example for the orbit transfer which is illustrated here, and "Infinitesimal" is different from in quantum mechanics for example, where many many things can happen in a second.

Hohmann transfer orbital manoeuver - all changes in velocity are induced by impulsive burns

Note: I'm not too sure "impulse" is an officially defined term just like force, work or energy - the only time I came across it was for rocket motors' specific impulse and control engineering. It might be used for convenience, as you may have noticed that an impulse has the same units as a change in linear momentum since Newtons are kg.m/s².

Suggestion: "Shock" may be what you're looking for, at least that's how spacecraft engineers call very high shortly applied forces like the mechanical wave produced by firing off explosive bolts.