Question

A car is moving with speed $30m/sec$ on a circular path of radius $500m$. Its speed is increasing at the rate of $2m/se{c}^2$. What is the net acceleration of the car at that moment (i.e the vector sum of tangential and radial acceleration)?

Answer 1

Velocity, $v=30$ $m/s$

Radius, $r=500$ $m$

Net acceleration is given by:

$\overrightarrow{a}=(\ddot{r}-r{\dot{\theta }}^2)\hat{r}+(r\ddot{\theta }+2\dot{r}\dot{\theta })\hat{\theta }$

where, $\hat{r}$ is radial direction,

$\hat{\theta }$ is tangential direction.

$\dot{r}=\ddot{r}=0$, Since, the radial distance is not changing. Also,

$\ddot{\theta }=2$ $m/s^2$

$\therefore \overrightarrow{a}=(0-500\times {\left(\frac{V}{r}\right)}^2)\hat{r}+(500\times 2+0)\hat{\theta }$

$⟹\overrightarrow{a}=-500\times \frac{900}{250000}\hat{r}+1000\hat{\theta }$

$⟹\overrightarrow{a}=-1.8\hat{r}+1000\hat{\theta }$

$|\overrightarrow{a}|=\sqrt{(1.8{)}^2+(1000{)}^2}$

$⟹|\overrightarrow{a}|=\sqrt{3.24+1000000}$

$⟹|\overrightarrow{a}|=1000.0017$ $m/s^2$