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If we knew the momentum of a moving object, how would we calculate the force exerted by that object upon impact with a surface?

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    $\begingroup$ If you know the rate of change in momentum, you can calculate the impact force $\endgroup$
    – Bob D
    Commented Oct 25, 2023 at 16:16
  • $\begingroup$ @BobD Wouldn't the rate of change in momentum just highlight the difference in force between the initial and final points, as opposed to the force itself? $\endgroup$
    – VV_721
    Commented Oct 25, 2023 at 16:54
  • $\begingroup$ You need a model of the force, as it depends on the details. The force will be a function of time. For instance, the dynamics and forces from a rubber ball and a billiard ball will be different. For a billiard ball you will have a larger force acting for a shorter time than for a rubber ball (smaller force acting along longer times), so that the total change in momentum is the same. $\endgroup$ Commented Oct 25, 2023 at 16:57
  • $\begingroup$ @VV_721 Rate of change in momentum tells us the force , an external force is responsible for this rate of change in momentum of the object. momentum in initial and final points wouldn't be force itself only the change in momentum for a short time yields a force $\endgroup$
    – Razz
    Commented Oct 25, 2023 at 17:08
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    $\begingroup$ @Razz Oh okay, I see what you're saying. I think I misunderstood your previous comment. When you mentioned rate of change in momentum, I thought you implied that the velocity of the object was changing as it approached the surface, but now I realize that the change in momentum is caused by the impact with the surface, which is what you meant. $\endgroup$
    – VV_721
    Commented Oct 25, 2023 at 17:22

2 Answers 2

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If you know the change in momentum of the object and the time interval over which this change takes place, then the average external force exerted on the object during this time interval is the change in momentum divided by the time interval. And, by Newton's Third Law, this is the same magnitude as the average force exerted by the object on whatever it impacts.

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As various people (@Bob D, @Pato Galmarini, @Razz) the force is the rate of change of the momentum. So, we need to not only know the momentum (before and after the collision with the surface) we also have to know how long the collision takes. In any semi-realistic situation the force will be varying with time, possibly in rather complicated ways, which I think is what @Pato Galmarini is hinting at by saying that we need to model the force.

Note that if you know the force applied to the moving object you know the force that the moving object applies to the surface, because of Newton's 3rd Law. The two forces have the same magnitude and point in opposite directions.

But without more information the question can't be answered. At minimum we would need the momentum of the object just before it hits the surface, the momentum just after it leaves the surface, and the amount of time it is in contact with the surface. That would not be enough information to find the force at any particular time, but it would allow us to find the average force. To find the force at any particular time we need some model for how the force depends on deformation of the object and the surface, and things become horrendously complicated.

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  • $\begingroup$ Horrendously complicated is an accurate description. I sat through a presentation describing instantaneous force modelling for the (semi)-elastic collision of two quartz spheres. The impulse forces were very non-linear as a function of time. $\endgroup$
    – Penguino
    Commented Oct 25, 2023 at 20:53
  • $\begingroup$ Yes @Penguino, I totally agree. Even if you do something really simple like model two particles colliding with an elastic Hooke's Law spring between them, a full analysis of the motion is much more difficult than one might at first imagine. With a semi-elastic collision between extended bodies like what you are describing I imagine the math to describe the local stresses must get truly awful! $\endgroup$ Commented Oct 25, 2023 at 21:24

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