Question

For work done problems, when do we include potential energy? I'm a bit confused about work done problems. When a cyclist cycles up a hill with different initial and final speed, his total work done would be 1/2 mv^2, regardless of mgh. What if he cycled up the hill with the same final and initial speed? When do we include mgh in work done problems? Or do we never include them in work done?

Answer 1

\[E = \Delta K + \Delta U \]\[K_{1}+U_{1} = K_{2}+U_{2}\]
Law of the conservation of mechanical energy
\[(1/2mk_{2}^2-1/2mk_{1}^2)+(mgh_{2}-mgh_{1})=0\]
If the equations are rearranged
\[1/2mk_{1}^2+mgh_{1} = 1/2mk_{2}^2+mgh_{2}\]

These problems really depend on the information given and the frame of reference.
I assume from the question you have given that the cyclists starts from rest. This is the only way that the work done is equal to the final kinetic energy.
\[W_{net} = \Delta K = \]\[K_{1}+U_{1} = 1/2mk_{2}^2-1/2mk_{1}^2\]\[W_{net} = \Delta K= 1/2mk_{2}^2-0=1/2mk_{2}^2\]
In this problem if the cyclist continues the same velocity up the entire hill the final work done is the \[\Delta K \]which is also the same as the change in \[\Delta U \]
To state it simply the work done is the change in kinetic energy. Once the biker reaches the top of the hill, this energy is stored as potential energy. The potential energy(potential to do work) is the same as the energy or work exerted to get him to the top of the hill.

Although the question states the total work done is the change in kinetic energy at the top of the hill, the change in kinetic energy is also equivalent to the potential energy at the top of the hill. Thus you can say using the conservation of Energy, that the total work applied due to kinetic energy is equal to the potential energy or the potential for gravity to do work on the biker if the biker goes back down the hill without peddling.

Answer 2

You can think of kinetic energy and potential energy in the analogy of a battery. You need to expend energy(do work) to charge a battery. Once the battery is charged, the max amount of energy(work) you can get out of the battery is equal to the amount of energy(work) used to charge it.

Answer 3

So is it safe to say that the change in KE that got him up the hill would equal to the PE he gained once he is up there?

Answer 4

Yes but don't double the work. Just know that if you were to add the change in kinetic energy and the change in potential energy they will be equal to zero. Later on you will cover work lost due to friction and other variables. But even in these situations you can account for these variables knowing that the total energy in a system is conserved. There are good instructional videos out there that explain this and other physics concepts. http://www.learner.org/resources/series42.html This link is to a series created by The Annenberg foundation. The videos are from the mid eighties, but the content is good. You can watch them on their site.