A dragster starts from rest and accelerates down a track. Each tire has a radius of 0.320 m and rolls without slipping. At a distance of 382 m, the angular speed of the wheels is 255 rad/s. Determine (a) the linear speed of the dragster and (b) the magnitude of the angular acceleration of its wheels.
V = omega r = 255 (.320)
382 = (1/2)V t
because distance = average speed time time if a is constant
solve that for t
t = 764 / V
382 = (1/2) a t^2
solve for a, LINEAR aceleration
alpha = angular acceleration = a/r
To determine the linear speed of the dragster, we can use the formula:
Linear speed = angular speed * radius
The angular speed is given as 255 rad/s, and the radius of each tire is 0.320 m. Substituting these values into the formula, we have:
Linear speed = 255 rad/s * 0.320 m = 81.6 m/s
Therefore, the linear speed of the dragster is 81.6 m/s.
To determine the magnitude of the angular acceleration of its wheels, we can use the equation:
Angular acceleration = (final angular speed - initial angular speed) / time
Since the dragster starts from rest, the initial angular speed is 0 rad/s. And we are not given the time it takes for the dragster to travel the given distance. However, we can use the information provided to calculate the time.
The distance traveled by the dragster is given as 382 m. The circumference of each tire can be calculated using the formula:
Circumference = 2 * π * radius
Substituting the radius value, we get:
Circumference = 2 * π * 0.320 m ≈ 2 m
Therefore, for every revolution of the tires, the dragster covers a distance of 2 meters.
To find out how many revolutions the tires have made, we can divide the distance traveled (382 m) by the distance covered in one revolution (2 m):
Number of revolutions = distance traveled / distance covered in one revolution = 382 m / 2 m = 191 revolutions
Since the tires roll without slipping, the number of revolutions is equal to the number of rotations made by the tires. Therefore, the final angular speed of the wheels can be calculated by multiplying the number of revolutions by 2π:
Final angular speed = number of revolutions * 2π = 191 revolutions * 2π ≈ 1201.3 rad/s
Now that we have the initial angular speed (0 rad/s), the final angular speed (1201.3 rad/s), and the distance traveled, we can substitute these values into the angular acceleration equation to get the magnitude of the angular acceleration.
Angular acceleration = (final angular speed - initial angular speed) / time
Solving for time:
time = distance traveled / linear speed = 382 m / 81.6 m/s ≈ 4.68 s
Substituting the values into the equation:
Angular acceleration = (1201.3 rad/s - 0 rad/s) / 4.68 s ≈ 257 rad/s^2
Therefore, the magnitude of the angular acceleration of the dragster's wheels is approximately 257 rad/s^2.