A race car is driven around a circular track at a constant speed of 180mph. If the diameter is 1/2 mile, what is the angular speed of the car? Express your answer in revolutions per hour.
The formula for angular speed is w=theta (in radians)/time
To find the angular speed of the race car, we first need to calculate the distance traveled by the car in one revolution around the circular track.
The formula to calculate the circumference of a circle is C = π * d, where C is the circumference and d is the diameter. Given that the diameter is 1/2 mile, the circumference is:
C = π * (1/2) = π/2 miles
Since the car is traveling at a constant speed of 180 mph, it covers the circumference of the track in one hour. This means the car completes one revolution per hour.
Now, let's convert the angular speed from revolutions per hour to radians per hour.
Since one revolution is equal to 2π radians, the angular speed can be calculated by dividing the number of revolutions per hour by 2π.
angular speed (in radians per hour) = 1 revolution / hour * 2π radians / 1 revolution
Simplifying, we find:
angular speed = 2π radians per hour
Therefore, the angular speed of the race car is 2π radians per hour, expressed in revolutions per hour.