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 heat transferred, m is the mass of the rotor, c is the specific heat capacity of the metal, and ΔT is the change in temperature.
In this case, the entire kinetic energy of the car is converted into thermal energy, which is then transferred to the brake rotors. The kinetic energy of an object is given by the equation:
KE = 1/2 mv^2
where KE is the kinetic energy, m is the mass, and v is the velocity.
Since the car stops completely, the initial kinetic energy is equal to the final thermal energy. Therefore:
1/2 mv^2 = mcΔT
Cancelling out the mass and rearranging the equation, we get:
v^2 = 2cΔT
Now we can solve for ΔT:
ΔT = v^2 / (2c)
Substituting the given values:
ΔT = (20.0 m/s)^2 / (2 × 420 J/(kg °C))
ΔT ≈ 0.238 °C
Therefore, the temperature of the brake rotors increases by approximately 0.238 degrees Celsius.