Question

A boy on a bicycle rides in a circle of radius ro at speed vo. If the boy now rides at a radius equal to half the initial radius ro, by what approximate factor must he change his speed in order to have the same radial acceleration

Answer 1

Answer:

The speed must change by $\frac{1}{\sqrt{2}}$ factor in order to have the same radial acceleration.

Explanation:

The radial or centripetal acceleration is given by:

$a_{c} = \frac{v^{2}}{r}$

Where:

v: is the speed = v₀

r: is the radius = r₀

$a_{c} = \frac{v_{0}^{2}}{r_{0}}$    (1)

If the radius is now equal to half the initial radius the speed must be:

$a_{c} = \frac{v^{2}}{r_{0}/2}$    (2)

By equating equation (1) and (2):

$\frac{v_{0}^{2}}{r_{0}} = \frac{v^{2}}{r_{0}/2}$  

$v^{2} = \frac{v_{0}^{2}}{2}$

$v = \frac{v_{0}}{\sqrt{2}}$      

Therefore, the speed must change by $\frac{1}{\sqrt{2}}$ factor in order to have the same radial acceleration.      

 

I hope it helps you!