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.[1] 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?
Answer
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.
No comments:
Post a Comment