A car with a mass of 890 kg travels at a speed of 20.0 m/s. Seeing that the road ahead is blocked by a rock slide, the driver applies the brakes, stopping the car. The work of stopping the car becomes thermal energy of the car’s four disk brake rotors; in other words, all of the energy goes into the brake rotors. Each rotor has a mass of 6.5 kg. The rotors are made of a metal with a specific heat capacity of 420 J/(kg °C).
After the car is stopped, how much has the temperature of its brake rotors increased? Give your answer in degrees Celsius. Assume that all four brake rotors receive the same amount of heat.
To find the temperature increase of the brake rotors, we can use the equation:
ΔQ = mcΔT
Where:
ΔQ is the amount of heat transferred to the rotors
m is the mass of the rotors
c is the specific heat capacity of the rotors
ΔT is the change in temperature
First, let's calculate the heat transferred to the rotors using the equation:
ΔQ = (mass of the car) * (initial velocity of the car)^2 / 2
Given:
mass of the car (m) = 890 kg
initial velocity of the car (v) = 20.0 m/s
ΔQ = (890 kg) * (20.0 m/s)^2 / 2
ΔQ = 890 kg * 400.0 m^2/s^2 / 2
ΔQ = 890 kg * 200.0 joules
ΔQ = 178,000 joules
Next, we need to divide the total heat transferred to the rotors equally among all four rotors:
Heat per rotor = ΔQ / 4
Heat per rotor = 178,000 joules / 4
Heat per rotor = 44,500 joules
Now, using the heat equation, we can calculate the change in temperature (ΔT) of each rotor:
ΔT = ΔQ / (m * c)
Given:
mass of each rotor (m) = 6.5 kg
specific heat capacity of the rotors (c) = 420 J/(kg °C)
ΔT = 44,500 joules / (6.5 kg * 420 J/(kg °C))
ΔT = 44,500 joules / (2730 J/°C)
ΔT = 16.32 °C
Therefore, the temperature of each brake rotor increases by approximately 16.32 degrees Celsius.