Question

The radii of the sprocket assemblies and the wheel of the bicycle in the figure are:
4 inches,
2 inches,
and 14 inches respectively.

If the cyclist is pedaling at a rate of 1 revolution per second, find the speed of the bicycle in (a) feet per second and (b) miles per hour.

Answer 1

To solve this task we have to make a proportion, but firstly we have to set up all the main points : so, the distance is  s=r(B), that has its r=radius,B=angle in rad velocity v=ds/dt= w(r)
Do not forget about  w = angular speed in rad/s and $w1 = 1 revolution/sec = 2Pi (rad/s)$
Now we can go to proportion
$v1=v2$
$w1*r1 = w2r2$$w2 = w1 * r1/r2 = 2w1 = 4Pi (rad/s)$
$w2 = w3 (which is the angular velocity of the rear wheel)$
SOLVING FOR A : $v3 = w3 * r3 = 4pi * 14 (inch/s) = 14.66 ft/sec$
$v3 = 14.66 ft/sec(1 mile/5280 ft)( 3600 sec/h)$$= 9.99$ or something about 10 mph --- SOLVING FOR B.
I'm sure it helps!