1. Time period of each car

Since both cars complete one full circle in the same time $t$, their time periods are equal: $T_1 = T_2 = t$

2. Speed in terms of radius and time period

For circular motion, $v = \frac{2\pi r}{T}$

So for the two cars: $v_1 = \frac{2\pi r_1}{t}, \qquad v_2 = \frac{2\pi r_2}{t}$

3. Centripetal acceleration formula

Centripetal acceleration is $a_c = \frac{v^2}{r}$

Thus, $a_1 = \frac{v_1^2}{r_1} = \frac{\left(\frac{2\pi r_1}{t}\right)^2}{r_1} = \frac{4\pi^2 r_1^2}{t^2 r_1} = \frac{4\pi^2 r_1}{t^2}$

Similarly, $a_2 = \frac{v_2^2}{r_2} = \frac{\left(\frac{2\pi r_2}{t}\right)^2}{r_2} = \frac{4\pi^2 r_2}{t^2}$

4. Take the ratio

$\frac{a_1}{a_2} = \frac{\frac{4\pi^2 r_1}{t^2}}{\frac{4\pi^2 r_2}{t^2}} = \frac{r_1}{r_2}$

So the ratio of centripetal accelerations is $a_1 : a_2 = r_1 : r_2$

5. Check options

• A: $m_1 r_1 : m_2 r_2$ ❌ Mass does not appear in centripetal acceleration.
• B: $m_1 : m_2$ ❌ Incorrect.
• C: $r_1 : r_2$ ✅ Correct.
• D: $1:1$ ❌ Only true if $r_1 = r_2$, not in general.

Therefore, the correct option is C.