Impulse Problem- time-average of normal force when running

[mtruong1999]

1. The problem statement, all variables, and given/known data
A somewhat paraphrased version of the problem:
Consider the vertical motion of the center of mass of a 150 lbs person as they run along a track. Suppose the person's cadence is 4 steps per second, and with every stride, they briefly leave the ground (thus the center of mass rises and falls repeatedly). Assume only one foot is in contact with the ground at a time, and this is brief contact. Thus the vertical (normal) force F(t) exerted by ground rises and falls in a sequence of short pulses, with an impulse occurring every 0.25 seconds.
a. Considering that only other vertical force acting on the runner is gravity, what must be the long-time time-weighted average of the vertical component of force of impact of F(t).
b. If the shoe is only in contact with the ground for 0.1 seconds, what is the short-time weighted average of F(t) over the 0.1 seconds?

Homework Equations

Average Force= (Δp)/(Δt)=(m*v_f - m*v_i)/Δt =
n= mg
Δp= change in momentum
n= normal force

The Attempt at a Solution

First, I converted 150 lbs into Newtons using 1 lb=4.45 N, just for the sake of units. 150lbs≅667.5N.
For part a, because we are considering the "long-time time-weighted average force", I immediately thought about taking a limit as time approaches infinity.
So, I took the limit as Δt→∞ of (mv_f - mv_i)/Δt, which turns out to be 0 because the numerator is a fixed value while the denominator increases without bound. Does this mean that the long time-weighted average of the normal force of impact is 0N?
I'm also a bit confused because my teacher is rather confusing and I have notes that I took from him where he set this limit equal to n-mg, thus resulting in 0=n-mg, why is this okay and legal to do??

For part b, I think I need the initial velocity, but its not given? I assumed that the velocity going down would be the same as the velocity of the foot leaving the ground, thus resulting in the formula Δp=2mv (and I would find the mass by dividing 667.5 N by acceleration of gravity), which I would then divide by 0.1 seconds to get the short-time weighted average. The only problem is: I don't know what the initial velocity is or any other way to get around needing a velocity.

 

[haruspex]

of (mvf - mvi)/Δt, which turns out to be 0

But there are multiple impacts over time, so in the limit infinitely many mv_f-mv_i terms.
Take a typical interval, representative of the long term process.