The formula for calculating angular speed is expressed as
w = v/r
where
w is the angular speed
v is the linear velocity
r is the radius of the wheel
We are considering the front tire. From the information given,
r = 19 inches
We would convert it to meters
Recall, 1 inch = 0.0254 m
19 inches = 19 x 0.0254 = 0.4826 m
v = 68 ft/s
We would convert 68 ft to meters
Recall, 1 foot = 0.3048 m
68 feet = 68 x 0.3048 = 20.7268
thus, v = 20.7268m/s
By substituting these values into the formula, we have
w = 20.7268/0.4826 = 42.95 rad/s
We need to convert to rpm. Let x rpm = 42.95 rad/s
1 rpm = 2pi/60 rad/s
Then,
x rpm = 42.95 rad/s
By crossmultiplying, we have
1 * 42.95 = x * 2pi/60
x = 42.95/2pi/60
If pi = 3.14, then
x = 42.95/2 * 3.14/60 = 42.95/0.105
x = 409 rpm
The angular speed of the front tire is 409 rpm