Question

An automobile has a mass of 1 405 kg, and its aluminum brakes have an overall mass of 6.45 kg. (Assume the melting point of aluminum is 660°C and the specific heat is 900 J/kg · °C.) (a) Assume all the mechanical energy that transforms into internal energy when the car stops is deposited in the brakes and no energy is transferred out of the brakes by heat. The brakes are originally at 21.0°C. How many times can the car be stopped from 27.0 m/s before the brakes start to melt?

Answer 1

Solution-

The energy is kinetic energy
E = 1/2 m v^2

When the braking is repeated N times, N times this energy is transferred into heat that will increase the aluminum's temperature.
The relation between heat energy change ΔQ and temperature change ΔT for a mass of aluminum M is

ΔQ = M * C * ΔT
where C is the heat capacity per unit mass ( taken as constant, 950 J / (kg C), over the temperature range in this problem)


Setting the heat energy needed for raising the temperature to the melting point to be less than the total kinetic energy provided in N braking experiments gives:

N > M C ΔT /(1/2 m v^2)

the melting temperature of Aluminum: it's about 660 C
So for the melting to start, the temperature increase we need is at least 660 C - 21 C = 639 C.
Therefore


N > 6.45 kg * 950 J/(kg C) * 639 C / (1/2 * 1 405 kg * (27.0 m/s)^2)
= 7.64

Hence N should be at least 8.