Question

A car rounds a curve at a steady 50 km/h. If it rounds the same curve at a steady 70 km/h, will its acceleration be any different? Explain.

Short answer

Yes, the acceleration will increase as the speed of the car increases.

Step-by-step solution

Step 1. Given Data

The initial speed of the car is $v_1=50\mathrm{km}/\mathrm{h}$.

The final speed of the car is $v_2=70\mathrm{km}/\mathrm{h}$.

Step 2. Understanding the centripetal acceleration

According to the relation of the centripetal acceleration, it is evident that centripetal acceleration gets affected by the change in speed of the vehicle and radius of the vehicle.

Step 3. Evaluating the variation in acceleration of the car

The relation of centripetal acceleration can be written as:

$a=\frac{v^2}{r}$

Here, v is the speed of the car and r is the radius.

The ratio of acceleration from the above relation is given by,

$\begin{array}{c}\frac{a_2}{a_1}=\left(\frac{\frac{{v_2}^2}{r}}{\frac{{v_1}^2}{r}}\right)\\ \frac{a_2}{a_1}=\left(\frac{{v_2}^2}{{v_1}^2}\right)\end{array}$

On plugging the values in the above relation.

$\begin{array}{c}\frac{a_2}{a_1}=\left(\frac{{\left(70\mathrm{km}/\mathrm{h}\right)}^2}{{\left(50\mathrm{km}/\mathrm{h}\right)}^2}\right)\\ \frac{a_2}{a_1}=1.96\\ a_2=1.96a_1\end{array}$

Thus, the acceleration will increase due to the increase in speed of the car.