How much energy is lost due to friction?

Question

A roller coaster's highest hill is always the first one. This is because friction pulls energy out of the coaster as it moves along a track. If the highest hill of a particular roller coaster is 16 m, and the if the coaster comes to rest at the top of the final hill which is 7 m tall. How much energy was lost due to friction along the way? The roller coaster has a mass of 172 kg.

Answer 1

In this problem we need to consider potential energy, kinetic energy, and work.  The initial potential energy plus the initial kinetic energy plus the work done on the system is equal to the final kinetic energy plus the final potential energy.

 

KE_i + PE_i + W_ext = KE_f + PE_f

 

In this problem we're assuming that the initial velocity (speed at the top of the first hill) is zero, so KE_i=(1/2)m(v_i)²=0.  Similarly, since it comes to a rest at the top of the final hill, KE_f is also zero.  You could still solve the problem if the coaster had a velocity at the top of the first hill, it's not a bad exercise to give that a try!

 

The potential energy is m*g*h, so PE_i = m*g*h_i and PE_f = m*g*h_f.  These are all known quantities.

 

Last we have the work done by external forces (in this case friction).  You need to be careful with the sign for work done by external forces.  In this case friction is removing energy from the system, so W_ext will be negative.  If we instead had an engine pushing the car along, that would be work adding energy to the system so instead it would be positive.

 

Now let's return to our original equation:

 

KE_i + PE_i + W_ext = KE_f + PE_f

0 + m*g*h_i - W = 0 + m*g*h_f

 

We want to solve for W

 

W = m*g*h_i - m*g*h_f 

W = (172 kg)*(9.8 m/s²)*(16 m) - (172 kg)*(9.8 m/s²)*(7 m)

W = 15,170.4 joules

 

So the system loses 15,170.4 joules of energy due to friction