A car is driven around a circular track. Which would have a greater effect on the magnitude

Was wondering if someone could help me with this question from my physics course. I'm taking it online, and the professor is not very responsive to emails asking for help. Any help would be appreciated. Thanks!!

A car is driven around a circular track. Which would have a greater effect on the magnitude of its acceleration, doubling the speed or moving to a track with half the radius? Please show your calculations.

Doubling the speed would quadruple the acceleration while halving the radius would only double the acceleration.

Centripetal acceleration is usually written as:

A = V² / R

We need to compare final acceleration (Af) with initial acceleration (Ai):

Af = (Vf)² / R <== Equation A
Ai = (Vi)² / R <== Equation B


If velocity is doubled, then Vf = (2 × Vi) and substituting into Equation A:
Af = (2 × Vi)² / R

Divide this by Equation B to get the ratio of the final acceleration to the initial acceleration:
= [(2 × Vi)² / R] / [(Vi)² / R]
= 4 × Vi² / Vi²
= 4




If radius, R, is halved, then Rf = Ri ÷ 2. The ratio of the final acceleration to initial acceleration is:

Af / Ai = [ V² / (Ri ÷ 2) ] / [ V² / (Ri) ]

= 2

Force = mass * tangential velocity^2 / radius

If you double the velocity force increases by 4 times

Half the radius would increase force by 2

Double velocity has a greater effect

Same argument applies to acceleration