A boy wants to jump onto a moving carousel that is spinning at a rate of 5 revolutions per minute. If the carousel is 64 ft in diameter, how fast must the boy run, in feet per second, to match the speed of the carousel and jump in? Round your answer to two decimal places.
n=5 rev/min = 5/60 rev/s
v=2πnR = πnD =3.14•5•64/60 =16.76 ft/s
To answer this question, we first need to determine the angular speed of the carousel. The angular speed is the rate at which the carousel rotates in radians per minute.
Given that the carousel spins at a rate of 5 revolutions per minute, we can convert this to radians per minute. Since there are 2π radians in one revolution, the angular speed is:
angular speed = 5 revolutions/minute × 2π radians/revolution
= 10π radians/minute
Next, we need to find the linear speed of the carousel. The linear speed is the distance traveled along the circumference in one minute.
The circumference of the carousel can be calculated using the formula C = πd, where d is the diameter of the carousel.
circumference = π × 64 ft
= 64π ft
Therefore, the linear speed of the carousel is equal to the circumference divided by the time taken to complete one revolution, which is 1 minute:
linear speed = 64π ft/minute
Finally, we convert this linear speed to feet per second by dividing by 60, since there are 60 seconds in a minute:
linear speed = 64π ft/minute ÷ 60
≈ 3.385 ft/second
So, the boy must run at a speed of approximately 3.39 feet per second to match the speed of the carousel and jump in.
To match the speed of the carousel and jump onto it, the boy needs to run at the same linear speed as the outer edge of the carousel.
Given that the carousel has a diameter of 64 ft, its radius (r) can be calculated by dividing the diameter by 2:
r = 64 ft / 2 = 32 ft.
The circumference of the carousel can be found by multiplying the radius by 2π (pi):
circumference = 2πr = 2 * 3.14 * 32 ft ≈ 201.06 ft.
Since the carousel is spinning at a rate of 5 revolutions per minute, the boy needs to match this speed by running in the same direction.
To calculate the linear speed at which the boy needs to run, we can multiply the circumference of the carousel by the rate of revolutions per minute:
linear speed = 201.06 ft/rev * 5 rev/min = 1005.3 ft/min.
To convert the linear speed from feet per minute to feet per second, we divide by 60 (since there are 60 seconds in a minute):
linear speed = 1005.3 ft/min / 60 s/min ≈ 16.76 ft/s.
Therefore, the boy needs to run at a speed of approximately 16.76 feet per second to match the speed of the carousel and jump onto it.