# The elusive difference between force and impulse

Impulse is defined as the product of a force $F$ acting for a (short) time $t$, $J = F*t$, and that is very clear. What I find difficult to understand is how a force can exist that doesn't act for a time.

If we consider the most common and observable force: gravity, the force of gravity is defined as $m*g$ and for a body of 1 Kg of mass is equivalent to $\approx$ 10 N.

But whenever we consider gravity we must consider the time, if a book falls from the table to the ground (h = .8m) the force acts for a (short) time t = 0.4 sec.

• Is there/can there be a force that doesn't act for a time?
• Can you explain why do not refer to the fall of the book as the impulse of gravity?
• Why if the same (short) time happens in a collision we call it an impulse?
• Isn't always a force actually an impulse?

update:

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. - John Rennie

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

• But, also when the book falls to the ground because of gravity there is a change of momentum, why is that not impulse? That is the elusive difference, for me.

force is not defined over a billion years, but:

a force is any interaction which tends to change the motion of an object. In other words, a force can cause an object with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to accelerate. Force can also be described by intuitive concepts such as a push or a pull.

therefore also in a collision there is a push on a ball, exactly the same as here: there is a push on the book that tends to change its motion. What is the difference?

## 4 Answers

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.

• @GreenRay: I'm not sure I fully grasp what you mean about the book, but I've extended my answer in the hope that it will help. – John Rennie Dec 23 '14 at 15:25
• @It helped indeed! You confirmed that whenever whatever force acts, it is an impulse we are applying. – user59485 Dec 25 '14 at 6:24

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.

• I hadn't seen that at the time of writing. Updated the answer with a couple more things, but physics is the same for books and satellites (not beyond that, where relativity takes over), though I agree it's easier to grasp with books. I thought it would be interesting to bring that in to illustrate. – Mister Mystère Dec 23 '14 at 17:53

"Isn't always a force actually an impulse?"

Force and impulse have different units--impulse has units of momentum, mass * distance/time (mass times velocity), while force has units of mass * distance/time^2 (mass times acceleration). If only a single force is acting on an object for a given time period, then multiplying the force by the time (or integrating the force over the time if the force isn't constant) gives the change in the object's momentum over that time, so in that case impulse is exactly equivalent to the change in momentum. (If multiple forces are acting over a time period, you have to perform vector addition on the impulses from each force to get the total change in momentum. That's why impulse is conceptually different than change in momentum--if a book is just resting on a table, over a given time period it will receive an impulse due to gravity and an impulse due to the normal force from the table, and these will cancel out so there is zero net change in the book's momentum.)

As an analogy, if an object is traveling at constant velocity for a given time period you can multiply the velocity by the time to get the change in position during that period, but this doesn't mean a velocity is "actually" just a change in position, since velocity is the rate that position changes with time. For example, two cars can both travel a distance of 60 miles (same change in position), but if one takes an hour to do it while the other takes two hours, their velocities are different. In the same way, two objects can both receive the same impulse, but if it took different time periods for them to receive that impulse, the forces on each object must have been different.

"Is there/can there be a force that doesn't act for a time?"

Realistically, all forces act over some time, although in some cases as contact forces during a collision, the force may act for only a tiny fraction of a second. In principle you could have a mathematical model of a force acting at only a single point in time but conveying a nonzero impulse (using a dirac delta function), but this wouldn't accurately describe any real-world situation we know of.

"Can you explain why do not refer to the fall of the book as the impulse of gravity?"

The change in momentum of the book could be described as an impulse due to gravity, but this is consistent with the book having a force of gravity acting on it at every moment during its fall.

"Why if the same (short) time happens in a collision we call it an impulse?"

Again, the impulse tells you the change in momentum, it's always understood that the change in momentum can be explained in terms of forces acting on both objects during the collision. In the case of an elastic collision (where none of the objects' linear kinetic energy is lost to heat) you can derive the change in momentum from conservation of kinetic energy and conservation of momentum, without ever needing to consider the forces. But you could analyze the forces during an elastic collision if you wished (this pdf has an analysis of forces during collisions of metal spheres), and you'd end up getting the same answer for the change in momentum as was arrived at just using conservation laws.

The force of gravity acting between the Earth and the Moon will be in effect for billions of years, until one of the bodies is destroyed.

For the purpose of many experiments, this can be considered to be indefinitely.