Getting the Smallest Possible Force

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Suppose that you are going to jump off a chair. Before you jump, let's think about the situation a little bit. When you land, you certainly don't want to hurt yourself, but you know that the floor has to exert a force on you to stop you. You want this force to be as small as possible, however. How do you arrange it?

First of all, once you jump you are in free fall. You are accelerating toward the floor at about 10 m/s²("g"), and your speed just before you hit the floor simply depends on the height of the chair. You can't hit the floor with less speed.

Your momentum just before you hit the floor equals your mass times your velocity. Your mass is going to be constant during the fall, and you can't adjust your velocity (since air resistance is not going to be a factor when you jump off a chair), so you can't lessen the momentum you have just before you hit the floor, either.

Once the floor stops you, your momentum will be zero. So your momentum is going to change from whatever it was just before you hit (which you can't control) to zero - you have no control over how much your momentum will change when you hit the floor.

Now, the impulse-momentum equation says that your change in momentum will equal the impulse that the floor exerts on you. Your change in momentum is determined by the height of the chair - you can't change that - so you also have no control over the amount of impulse that the floor will exert on you. Once you left the chair, the impulse that the floor will exert on you was fixed.

It is beginning to look like you really don't have a lot of control over this situation, doesn't it? (This might explain why people get hurt falling down, huh?) The impulse that the floor exerts on you to stop you is determined by your mass and the height of your jump, and nothing else.

However, the impulse that the floor exerts on you depends on two things - the force that the floor exerts and the time that it exerts it. You DO have some control over the stopping time! Suppose that you flex your knees when you land. This increases the time that the floor stops you, thereby decreasing the force that the floor has to exert. Most people flex their knees instinctively in this situation (or maybe they tried it with locked knees and got hurt!).

By doubling the stopping time, the floor can exert the same impulse on you with half the force. With five times the stopping time, the floor has to exert just one-fifth of the force to exert the same impulse on you.

The same principles explain why it is better to fall on a soft carpet or mat than on a hard floor - the stopping time is automatically increased, thereby decreasing the stopping force.

In summary:

You can't change your velocity just before you hit the floor.
So you can't change your momentum just before you hit the floor.
So you can't change the amount your momentum needs to change when the floor stops you.
Since the impulse that the floor has to exert to stop you equals your change in momentum, the floor has to exert the same impulse on you no matter how you land.
But, since impulse equals force times time, you can decrease the force that the floor exerts on you by increasing the stopping time, by bending your knees or landing on a soft mat (or both).

last update April 11, 2000 by JL Stanbrough