An automobile has a mass of 1500 kg, and its aluminum brakes have an overall mas

Question

An automobile has a mass of 1500 kg, and its aluminum brakes have an overall mass of 6.00 kg. (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 20.0°C. What will be the temperature of the brakes when the car is stopped from 30.0 m/s.

Explanation / Answer

First ask yourself: what is the energy of a car of mass m moving at a speed v? Of course that's the kinetic energy E = 1/2 m v^2 In the situation in (a), 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 ?Q and temperature change ?T for a mass of aluminum M is ?Q = M * C * ?T 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 ?T /(1/2 m v^2) Now we need to know 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 - 20 C = 640 C. Therefore N > 6.00 kg * 950 J/(kg C) * 640 C / (1/2 * 1500 kg * (30 m/s)^2) = 7.78 So N should be at least 8. (b) A number of effects we ignored: - energy is lost through friction with the road and air - energy is lost through heat loss - the brakes are not made of pure Al, but we did take C and the melting point of pure Aluminum. - The heat capacity is not constant over such a temperature range, being bigger at higher temperatures. We took an average here.

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