When you climb a flight of stairs, you are moving your body weight (a
force) up a certain distance (the vertical height from the bottom to the
top stair). In which case does the amount of work you do increase?
Explain your answer.
a. You run up 10 stairs then you run up 50 stairs.
b. You walk slowly up 10 stairs and then you run up 10 stairs really fast.

1 - You run up 10 stairs then you run up 50 stairs.

Explanation:

You have the following cases:

1 - You run up 10 stairs then you run up 50 stairs.

2 - You walk slowly up 10 stairs and then you run up 10 stairs really fast.

The amount of work done is higher in the first case, because the work only depends of the vertical displacement and not of the time. In this situation the work is agains the gravitational force, that always points directly downward.

In the first case you traveled a vertical distance higher than in the second case. In the second case you traveled certain distance in a lower time, but work does not depend of this parameter. When you want you calculate the work done per unit of time you calculate the power.