Question

When brett and will ride the​ carousel, brett always selects a horse on the outside​ row, whereas will prefers the row closest to the center. these rows are 19 ft 5 in. and 12 ft 3 in. from the​ center, respectively. the angular speed of the carousel is 2.2 revolutions per minute. what is the​ difference, in miles per​ hour, in the linear speeds of brett and​ will?

Answer 1

When the motion is traveling on a straight line, we say that it travels at a certain linear speed or velocity. But when it is in circular motion, the distance it travels is angular. So, this time, we say that it travels at a certain angular speed or velocity. The relationship between linear velocity and angular velocity is:

V = rω
where
V is the linear velocity
r is the radius of the circle
ω is the angular velocity

The final answer should be in miles per hour. To be consistent, let's transform all distances to miles and time to hour.

Brett's distance:
19 ft + 12 in = 19 ft + 12 in (1 ft/12 in) = 20 ft
1 foot is equal to 0.000189394 mile
20 ft * 0.000189394 = 0.00378788 miles

Will's distance:
12 ft + 3 in = 12 ft + 3(1ft/12in) = 12.25 ft
12.25 ft * 0.000189394 = 0.00232 miles

Next, we convert ω from revolution per minute to radians per minute. One revolution is equal to 2π radians, and 60 minutes is equal to 1 hour:

ω = 2.2 rev/min * (2π rad/1 rev) * (60 min/1 hr)
ω = 264π rad/hour

Now, we can determine the linear velocities of Brett and Will:

Brett's linear velocity = (0.00378788 miles)(264π rad/hour)
Brett's linear velocity = 3.142 miles/h

Will's linear velocity = (0.00232 miles)(264π rad/hour)
Will's linear velocity = 1.924 miles/h