The average car today has a mass of 1100 kg, and when accelerating from rest, covers 0.25 miles in 15 seconds. Each rim and tire together has a diameter of 46 cm and a mass of 9.1kg. If we agree the rim and tire have the shape of a solid disk that rotates through its geometric center, what would be the kinetic energy of one of the tires, in Joules, at the end of the run?

To find the kinetic energy of one of the tires at the end of the run, we need to calculate the linear and rotational kinetic energies separately, and then add them together.

First, let's calculate the linear kinetic energy:

1. Find the final velocity of the car:
To do this, we need to convert the distance traveled from miles to meters:
Distance = 0.25 miles = 0.25 * 1609.34 meters
= 402.34 meters

Then, we can use the formula for average velocity:
Velocity = Distance / Time = 402.34 meters / 15 seconds
= 26.82 m/s

2. Calculate the linear kinetic energy:
The formula for linear kinetic energy is:
Kinetic energy = 0.5 * mass * velocity^2

Mass of the car = 1100 kg

Kinetic energy = 0.5 * 1100 kg * (26.82 m/s)^2
= 399,380.18 Joules

Next, let's calculate the rotational kinetic energy of the tire:

1. Find the moment of inertia (I) of the tire:
The moment of inertia for a solid disk rotating about its geometric center is given by the formula:
Moment of inertia (I) = 0.5 * mass * radius^2

Mass of the tire = 9.1 kg

Since the diameter of the tire is given, we need to convert it to radius:
Radius = Diameter / 2 = 46 cm / 2 = 23 cm = 0.23 meters

Moment of inertia (I) = 0.5 * 9.1 kg * (0.23 meters)^2
= 0.2355275 kg·m^2

2. Calculate the rotational kinetic energy:
The formula for rotational kinetic energy is:
Rotational kinetic energy = 0.5 * moment of inertia (I) * angular velocity^2

Since the tire is rotating through its geometric center, its angular velocity is equal to the linear velocity divided by the radius:
Angular velocity = Velocity / Radius = 26.82 m/s / 0.23 meters
= 116.78 rad/s

Rotational kinetic energy = 0.5 * 0.2355275 kg·m^2 * (116.78 rad/s)^2
= 1582.41 Joules

Finally, to find the total kinetic energy, we add the linear and rotational kinetic energies together:

Total kinetic energy = Linear kinetic energy + Rotational kinetic energy
= 399,380.18 Joules + 1582.41 Joules
≈ 400,962.59 Joules

Therefore, the kinetic energy of one of the tires at the end of the run is approximately 400,962.59 Joules.