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When you apply the work energy theorem i.e.

$$ \int F(x) dx = \Delta E_{\mathrm{kin}} $$

you typically know the initial speed and the force distance relation $F(x)$ and then wants to calculate the final speed. For example in calculating the speed of an arrow of a compound bow. There are many examples like that.

The "impulse momentum" theorem (i.e. newtons second law in integral form) has a similar mathematical structure but integrating over time instead:

$$ \int F(t) d t = \Delta p $$

What are (real world) examples where you know $F(t)$ (analytically or just graphically) and want to calculate $\Delta p$ (or $\Delta v$)?

The only example I found so far was to calculate the take off speed in jumping in biomechanics there you measure $F(t)$ by using a force platform. However I don't see in this case why not measuring the take of speed directly (using a light barrier, or using video analysis, with a slow motion camera or just calculating it from flight time or height), so the example doesn't really count.

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When you say "I don't see in this case why not measuring the take of speed directly" you seem to assume several things.

  1. Mass is known
  2. Mass is constant
  3. Velocity is clearly defined
  4. Maybe you are even thinking only of rigid bodies

If any of these conditions fail, you can construct an example yourself.

-> 1. momentum conservation holds independent of whether you know the mass of the object; often you first conclude on the momentum, and from there you conclude on the mass; if you measure velocity, you only have half the information to relate it to momentum conservation

-> 2. for a rocket (or in special relativity) mass is not even constant, but momentum conservation remains true nonetheless, if properly applied to the rocket and the ejected fuel

-> 3. if a mechanical system is hit by a force, generally everything starts to move relative to each other, so how do you properly measure velocity? the answer is of course: velocity of the center of gravity (and this is already a consequence of momentum conservation); but the latter might not be easy to determine just with a light barrier or something on the table; think of a big glass of Mai Tai with ice cubes and all, and you push it, then the glass moves, the drink moves, the ice cubes move...

-> 4. related to 3, but more specifically with respect to solids: not every solid system is automatically rigid; most technical systems are elastic internally; a car for example is suspended over wheels, springs, control arms; and if it gets a bump from the road, what does that mean for vertical momentum of the car during that process, where it starts vibrating all over? "impulse momentum theorem" tells it to you, if you have measured the force

I think biomechanics is actually generally a very good example because the human body consists of bones, joints, soft tissue, internal forces (muscles) and even a control system (brain). Treating it as if it was just like a dumb billiards ball is clearly not effective.

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