Why is it the same amount of work for a hiker to climb straight up a cliff than to walk up a slope?

Answer 1

Work equals force times distance. Climbing a cliff takes more force, but the distance is much shorter. The inverse applies to the slope. Mathematically:

d = (1/sinz)*h where h is the height and z is the angle from start point to end.

F= ma where m is constant. a is a sum of several vectors, but the most important ones are gravity and the forward acceleration.

If fighting gravity directly, a = (ac) - 9.8 where (ac) is the acceleration of the climber.

If going up a slope, a = <x,y>-<0,9.8>, where x and y are the forward and upward acceleration of the climber.

With all that, just solve for W = m*a*d, or W = m(h(sqrt(x^2+(y-9.8)^2)/sinz).

Note that when x= 0, W= (mh(y-9.8))

And when x =/= 0, W = m(h(sinz(y-9.8)/sinz)) = mh(y-9.8)

Answer 2

While the distance traveled may be different in each scenario, the work done in each case is the same because work is a function of force applied over a distance, not the path taken. Both scenarios require the hiker to exert the same amount of force against gravity to reach the same vertical height.