1) In classical mechanics, impulse is the product of a force, F, and the time, t, for which it acts. The impulse of a force acting for a given time interval is equal to the change in linear momentum produced over that interval...($t_1-t_2$).The SI unit of impulse is the newton >second (N·s) or, in base units, the kilogram meter per second (kg**·m/s)**.
2) A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly.Therefore
$\begin{align} \mathbf{J} &= \int_{t_1}^{t_2} \frac{d\mathbf{p}}{dt}\, dt =\Delta \mathbf{p} \end{align}$
I read a couple of previous answers, but that hasn't helped me understand: one repeats the above definition, the second says: "In the Newtonian point of view, impulse and change of momentum are different concepts...."
wiki's definition is as confusing: $J = \Delta p$
In order to simplify most, let's consider the unitary mass (m = 1) since velocity is what we call the momentum of unitary mass. Impulse is equal to the change of momentum/velocity/ : $J =[1*] \Delta v$ ( * sec)?
Summing up, considering m=1, we have:
- Velocity = $v =[1*] v$ (m) $v$
- Momentum = $p =[1*] v$ (m) $v$
- Acceleration = $a = [1] \Delta v$ change of velocity (m) $v/s$
- Force = $F =[1*] \Delta v$ change of momentum (m) $v/s$
- Impulse of a force = $J =[1*] \Delta v$ change of momentum, (m) $ = v$
Unless I made some mistakes, impulse is equal to momentum and not to change of momentum. Where did I go wrong, or, what is the final word?
2) as to the second period, I thought that the proportion between longer time and bigger change is valid only if the force gives constant acceleration, like gravity. How can that definition apply to collisions, where a ball gives a fixed amount of momentum which cannot be increased by duration? A cue ball hitting another ball gives constant acceleration? A bowler throwing a bowl on a lane gives constant acceleration? Does it matter if his arm swings for 1 or two seconds?.
I am confused and making confusion. Can you clarify my doubts?
update:
Your problem is that acceleration isn't the change in velocity
what is change of velocity then? if a football is at rest and I kick it and it aquires v=10m/s, haven't I accelerated it over a period of time? isn't that difference of $\Delta v= +10 m/s$ acceleration?,
but (taking there is a mistake there) my question was not about acceleration but:
. Unless I made some mistakes, impulse is equal to momentum and not to change of momentum. what is the final word?
- is change of momentum/velocity the same as momentum/velocity? how can they have the same units?
that statement applies to forces that can be sustained over some time.
I said: consider the hand of a bowler, it pushes a bowl with a force. If he pushes it for half a second or a second the change of momentum is the same, what changes is only that he can aim at the target with more precision