Difference in average force by using different formulas

My textbook has an exercise in which it leads you to calculate the average value of a force with non-uniform magnitude by the second law, $$F=\frac{\Delta p}{\Delta t}$$, then calculating the distance traveled by the area of a force in function of time graph and using it to calculate the average force again via the kinetic energy theorem, $$F \cdot \Delta s=\Delta E_k \rightarrow F=\frac{\Delta E_k}{\Delta s}$$. This leads to different values, and that is the purpose of the exercise, but why exactly is that so and what does each value mean? I guess it happens because if I'm not mistaken (my calculus understanding is very basic) the kinetic energy is the integral of force in respect to distance and momentum is the integral of force in respect to time, but I don't actually get what this means.

1 Answer

I think your hunch is on the right track.

In the first case you have $$\bar{F}_1=\frac{\Delta p}{\Delta t}=\frac{1}{t_2-t_1}\int_{t_1}^{t_2} dp=\frac{1}{t_2-t_1}\int_{t_1}^{t_2} \frac{dp}{dt}dt=\frac{1}{t_2-t_1}\int_{t_1}^{t_2} F(t)dt$$ In the second case $$\bar{F}_2=\frac{\Delta E_k}{\Delta s}=\frac{1}{s_2-s_1}\int_{s_1}^{s_2} dE_k=\frac{1}{s_2-s_1}\int_{s_1}^{s_2} \frac{dE_k}{ds}ds=\frac{1}{s_2-s_1}\int_{s_1}^{s_2}F(s)ds.$$ So in the first case you are computing average force over time, while in the second case over the distance, which is not the same thing.