Real-Life Math -- Solution
Part 1:
m = the total mass of the brakes in kg
m = 4 brakes x 5 kg per brake
c = 900 joules/kg degree C
dT = the change in temperature
dT = 660 C - 20 C
Q = m c dT
Q = (4 x5 kg) x (900 joules/kg degree C) x (640 degrees C)
Q = 11,520,000 joules
Q = approximately 12 million joules
Part 2:
Since PE = Q, we know that PE = 12,000,000 joules
m = the mass of the car
m = 1,500 kg
g = 9.8 m/s2 is the gravitational acceleration constant
PE = mgh
12,000,000 joules = (1,500 kg) x (9.8m/s2) x h
12,000,000 joules = (14,700 kg m/s2) x h
h = 816 m
That's about 0.8 km.
Part 3:
The grade is 1/10.
By cross-multiplying, we find:
x = 0.8 x 10
x = 8 km
This means that the brakes would start to melt after the car had gone a distance of 8 kilometers down the road. Although this is just an estimate, 8 kilometers is not an uncommon length for mountain roads, so overheating is a potential problem.
An engineer would thus pursue this problem further, taking into account 2 important effects we have neglected: the brake disks would cool due to air flow over the brakes, and the temperature would not be uniform in the disks but would be higher on the braking surface.