Problem 4: The roller coaster car starts its journey from a vertical height of 40 m on the first hill and reaches a vertical height of only 25 m on the second hill, where it slows to a momentary stop. It travelled a total distance of 400 m. Determine the thermal energy produced and estimate the average friction force on the car whose mass is 1000 kg.

Solution

Theory: The transformations of K.E and P.E are not always perfect in a roller coaster car. There are a number of frictions like air, or between the steel wheels of the car and the track. Due to this the K.E mechanical energy is continuously dissipated as heat, sound energies etc. This is the reason why the second hill is less tall than the first one. Note that the friction is mechanically set to increase where the car comes to a stop.

Therefore, the energy at the top of first hill is greater than the energy possessed at the top of the second hill. The difference in energy is assumed to have transferred to heat.

## What is a roller coaster?

A roller coaster car is a mechanism that uses gravity and inertia to send a car on a winding track (i-e, track with a number of hills, bottoms and curves). Due to gravity and inertia, energy of the car is continuously exchanged between gravitational potential energy and kinetic energy and the car moves.

Initially, the car is moved to the top of the hill (hill is a place of high elevation on the track of the car) with the help of a chain. Here the car has some P.E due to its elevated location. This energy is converted to K.E as the car moves down from the hill to the bottom on the track. The speedily moving car at the bottom moves up on another (second) hill where it gains P.E once again. That energy is converted to the K.E as the car comes down from the hill to the bottom. Here it goes up a third hill and the process goes on.

If the track consists of turns, friction between the wheels of the car and track is utilized to provide the necessary centripetal force for the curve.

Given

Height of the first hill = 40 m

Height of second hill = 25 m

Mass of the car = 1000 kg

Distance travelled = 400 m

(a) Thermal Energy (b) Average force of friction

(a) P.E possessed by the car at the first hill = mgh = 1000 × 9.8 × 40 = 392000 J
When the car moves down this energy is converted to K.E till it is zero at the bottom. The K.E is then converted to P.E when the car elevates to the second hill. However, some of the K.E in the whole journey is also converted to heat or thermal energy which heats up the tyres and track of the car.
P.E possessed by the car at the second hill = mgh = 1000 × 9.8 × 25 = 245000 J
This energy is less than the initial energy. The difference in energies is the energy converted to heat.
Therefore, difference in energy,
39200 – 245000 = 147000 J is the required heat energy.

(b) Now from work energy theorem, ΔK.E = Work = Fr.d ⇒Fr= (ΔK.E)/d. So put the values,
Fr = 147000/400 = 368 N