Once he hits the ground, the man quickly comes to rest. That is, his momentum changes from 540 kg·m/s to
0.
If the man does not bend his knees, then
The negative sign in our answer just means that the force exerted on the man is directed in the negative
direction: up.
Now, what if he had bent his knees?
If he bends his knees, he allows for his momentum to change more slowly, and as a result, the ground
exerts a lot less force on him than had he landed stiff-legged. More to the point, hundreds of thousands of
newtons applied to a person’s legs will cause major damage—this is the equivalent of almost 20 tons
sitting on his legs. So we would assume that the man would bend his knees upon landing, reducing the
force on his legs by a factor of 30.
Calculus Version of the Impulse–Momentum Theorem
Conceptually, you should think of impulse as change in momentum, also equal to a force multiplied by the
time during which that force acts. This is sufficient when the force in question is constant, or when you
can easily define an average force during a time interval.
But what about when a force is changing with time? The relationship between force and momentum in
the language of calculus is
A common AP question, then, gives momentum of an object as a function of time, and asks you to take the
derivative to find the force on the object.
It’s also useful to understand this calculus graphically. Given a graph of momentum vs. time, the slope
of the tangent to the graph gives the force at that point in time. Given a graph of force vs. time, the area
under that graph is impulse, or change in momentum during that time interval.