Real-Life Math
Math is used in most areas of a physicist's work.
"Excellent mathematical skills are a must -- even for experimentalists," says Robert Ragan. He is a research physicist. "Take your math classes seriously."
This is a basic math problem required in an introductory physics course at the university level.
It is a problem that automobile engineers have to think about: cooling of brakes. For example, the brakes may overheat if a driver rides the brakes while driving down a long steep road.
Calculate the distance a car can go before the brakes start to melt, if a 1,500-kg automobile travels down a steep mountain with a grade of 1/10 and the driver maintains a constant speed by holding down the brakes.
Part 1:
Assume the brakes are made of aluminum, with each disk weighing 5 kg. The amount of heat energy needed to raise the temperature of the brakes from 20 C to their melting temperature of 660 C is the first thing that we need to calculate. This value is called Q.
Q = m c dT
m = the total mass of the brakes in kg
c = 900 joules/kg degrees C is the specific heat of aluminum
dT = delta T or the change in temperature
Calculate Q.
Part 2:
As the car travels down the hill, gravitational potential energy (PE) is converted into kinetic energy, which is then converted into heat at a constant rate by the friction in the brakes. Since the potential energy is converted into heat energy, we can set PE = Q. For a change in elevation of h meters, the amount of potential energy that is converted into heat is:
PE = mgh
m = the mass of the car
g = 9.8 m/s2 is the gravitational acceleration constant
h = the change in elevation necessary to start melting the brakes
Solve for h.
Part 3:
Since we are told that the grade or slope of the road is 1/10, we know that for every 1 vertical kilometer, a car will travel 10 kilometers along the road. Find how far down the road the car will go before starting to melt its brakes.