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## Homework Statement

Two identical twins, A & B, are riding identical bikes up the same hill, both at constant speed. Twin A takes 20 seconds to climb the hill, while twin B takes 40 seconds.

a) Neglecting all forms of friction, which twin consumes more energy?

b) With friction, which twin consumes more energy?

## Homework Equations

Kinetic E = ½mv[2]

Gravitational Potential E = mgh

## The Attempt at a Solution

a) Intuitively, I know that both consume the same amount of energy.

Their gain in potential energy is the same, as they are going up the same hill so h, or height, must be the sa

me.

However, I do not understand how to explain it using the equations. Twin A would have 4x the kinetic energy of Twin B, so the kinetic E applied does not equal the potential E gained. (½mv[2] ≠ mgh).

So, should I instead think of the total energy of each biker?:

Total E consumed = ½mv[2] + mgh

b) The biker going faster (Twin A) would consume more energy. I think that with a greater velocity they will encounter greater friction. (Is this correct?) My professor said friction is just random or nondirectional kinetic energy. How can I explain this mathematically with just the equations given?