Impact Force vs Newton’s Second Law – Is There a Difference Between Impact Force and Force from Newton’s Second Law?

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So me and my friend were having a discussion today that why should our legs break if we jump off from a taller height like 20 meters than if we just jump from a mere meter. We both agreed that our legs should break from a taller height, but we couldn't determine the reason behind it. I mean it doesn't matter if I drop from 20 meters or only 1 meter, the acceleration due to gravity will be same in both cases, right? So from Newton's second law, i.e. $F=ma$, my weight should be the same regardless of from which height I jump. Now, my weight is not gonna change even after hitting the ground, so shouldn't the reaction force that I get from the ground be equal to my weight since that is the force which I am applying on the ground during impact?

If no, then why not? Moreover, if the impact force is not the same as my weight (presumably higher than it), then why do I only apply my weight as a force on the ground when I stand on it, but the force becomes different on impact?

Best Answer

The normal force from the ground must be large enough to give you an upward acceleration, that is large enough so that you don't melt into the ground. First of all, this means that the normal force will be time dependent: while you are in the air, the normal force is zero; then, as you hit the ground, the normal force grows very rapidly, causing acceleration so that you slow down; finally, the normal force reduces again so that when your velocity is zero, the normal force is equal to you weight, so that you remain on the ground.

When you are jumping from 1 meter, your velocity when you hit the ground is small, and the acceleration needed to make you stand still is small. Hence the maximum normal force does not need to be extremely large.

When you are jumping from 20 meters, your velocity when you hit the ground is large, and the acceleration needed to make you stand still is large. Hence the maximum normal force needs to be large*.

*Actually, the impulse must be large, which is the time integral of the normal force over the period of landing, from first contact to stand-still. This impulse is equal to the change in momentum.

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