# Work With Regards to Velocity

## Main Question or Discussion Point

You'll have to excuse me if this issue has been discussed here before. I did about 30 minutes worth of searching and didn't come upon it, so I decided to post.

A colleague and I had a lengthy discussion today on whether running a set distance spends more calories (energy) than walking that same distance.

My main argument centered around moving a mass over a distance.

My contention is that work is the product of force and distance:

W = F * d

If the same mass, m, is moved across the same distance, d, then the same force, F, is exerted. The hinge point of my argument is the assumption that your acceleration/deceleration is instantaneous.

If you look at the free body diagram, you'll note that you have force acting in two axis on the mass.

In the x-axis, you have the applied force, F, and the friction force, F_f. According to Newton's Second Law, a body in motion will stay in motion unless acted upon. Hence, in order to keep a constant velocity,

F = F_f hence $$\Sigma$$ F_x=0

In the y-axis, there's a gravitational force,

F_g = m * g

and the equal and opposite normal force, F_N

F_g = F_N hence $$\Sigma$$ F_y=0

The kinematic coefficient of fricition, u, is the same, regardless of velocity. The equation for friction force is:

F_f = F_N * u * cos(theta) <-- Theta being the angle at which the plane the mass is on is at. Assume a flat plain (cos0 = 1) for simplicity here.

Regardless of velocity, u and F_N are the same, hence F_f is the same.

So, if F_f is the same, F is the same. And, in the end,

W = F * d produces the same product regardless of velocity.

In the end, I convinced my coworker that my approach was correct. Although, he didn't buy the end result.

I'd love to hear some of your thoughts on this subject.