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 need to calculate the amount of heat absorbed by the rotors during the braking process. The equation for heat (Q) is given by:

Q = mcΔT

Where Q is the heat absorbed, m is the mass of the object, c is the specific heat capacity of the object, and ΔT is the change in temperature.

Given:
Mass of each brake rotor (m) = 6.5 kg
Specific heat capacity of the metal (c) = 420 J/(kg °C)

First, we need to calculate the total heat absorbed by the brake rotors. Since all four rotors receive the same amount of heat, we can multiply the heat absorbed by one rotor by four to get the total heat.

Q_total = Q_rotor x 4

Next, let's calculate the heat absorbed by one rotor (Q_rotor).
To do this, we need to find the change in temperature (ΔT) of the rotor. The change in temperature can be calculated using the equation:

ΔT = Q_rotor / (mc)

Rearranging the equation, we get:

Q_rotor = ΔT x mc

Now, we substitute the given values into the equation:

m = 6.5 kg (mass of each rotor)
c = 420 J/(kg °C) (specific heat capacity)
Q_rotor = ? (heat absorbed by one rotor)

We know that the mass of the car (890 kg) gets converted into heat in the rotors, so the total heat absorbed by the rotors should be equal to the initial kinetic energy of the car.

Initial kinetic energy of the car (KE) = 1/2 mv^2

Substituting the given values:

KE = 1/2 x 890 kg x (20.0 m/s)^2

Now, we can equate the total heat absorbed by the rotors to the initial kinetic energy of the car:

Q_total = KE = Q_rotor x 4

Substituting the values into the equation:

Q_rotor x 4 = 1/2 x 890 kg x (20.0 m/s)^2

Simplifying the equation, we can solve for Q_rotor:

Q_rotor = (1/2 x 890 kg x (20.0 m/s)^2) / 4

Now that we have Q_rotor, we can use the equation ΔT = Q_rotor / (mc) to find the change in temperature of each rotor.

Substituting the given values into the equation:

ΔT = (Q_rotor) / (mc) = (Q_rotor) / (6.5 kg x 420 J/(kg °C))

Calculating the value:

ΔT = (Q_rotor) / (6.5 kg x 420 J/(kg °C))

Finally, we can solve for ΔT to find the change in temperature of each rotor.