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 calculate the kinetic energy of one tire at the end of the run, we need to consider both the linear and rotational motion of the tire.
First, let's calculate the linear speed of the car. We know that it covers 0.25 miles in 15 seconds. To convert miles to meters, we multiply by 1609.34 (since 1 mile is approximately 1609.34 meters):
0.25 miles * 1609.34 meters/mile = 402.335 meters
The linear speed is then calculated by dividing the distance traveled by the time taken:
Linear speed = Distance / Time = 402.335 meters / 15 seconds = 26.822 m/s
Next, let's calculate the rotational speed of the tire. The tire has a diameter of 46 cm, which means the radius is half of that, or 23 cm (0.23 meters). To convert this to the circumference, we multiply by 2π:
Circumference = 2π * Radius = 2π * 0.23 meters ≈ 1.445 meters
The rotational speed can be calculated by dividing the circumference by the time taken:
Rotational speed = Circumference / Time = 1.445 meters / 15 seconds ≈ 0.0964 m/s
Now that we have the linear speed and rotational speed, we can calculate the total speed of the tire. Since the tire rotates through its geometric center, the linear and rotational speeds are additive:
Total speed = Linear speed + Rotational speed = 26.822 m/s + 0.0964 m/s = 26.9184 m/s
Finally, we can calculate the kinetic energy of the tire using the formula:
Kinetic energy = (1/2) * Mass * Velocity^2
The mass of the tire is given as 9.1 kg. Substituting the values into the formula:
Kinetic energy = (1/2) * 9.1 kg * (26.9184 m/s)^2 ≈ 6,768.04 Joules
Therefore, the kinetic energy of one tire at the end of the run is approximately 6,768.04 Joules.