Question

A car starts from rest on a curve with a radius of $120\mathrm{m}$and accelerates tangentially at $1.0\mathrm{m}/{\mathrm{s}}^2$ . Through what angle will the car have traveled when the magnitude of its total acceleration is $2.0\mathrm{m}/{\mathrm{s}}^2$ ?

Short answer

$\mathrm{\theta }=49.64°$

Step-by-step solution

Given Information

We are given with:

$$\begin{gathered} \mathrm{radius}=\mathrm{r}=120\mathrm{m} \\ \mathrm{tangential}\mathrm{acceleration}={\mathrm{a}}_{\mathrm{t}}=1\mathrm{m}/{\mathrm{s}}^2 \\ \mathrm{total}\mathrm{acceleration}=\mathrm{a}=2\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Calculation

The car have traveled when the magnitude of its total acceleration is $\mathrm{a}=2\mathrm{m}/{\mathrm{s}}^2$.

As we know the formula for angle:

$$\begin{gathered} \mathrm{\theta }=\frac{\mathrm{d}}{\mathrm{r}} \\ \mathrm{d}=\mathrm{distance} \\ \mathrm{r}=\mathrm{radius} \end{gathered}$$

For distance:

localid="1648908492211" $$\begin{gathered} \mathrm{d}=\frac{1}{2}{\mathrm{a}}_{\mathrm{t}}{\mathrm{t}}^2 \\ \mathrm{t}=\frac{\mathrm{v}}{{\mathrm{a}}_{\mathrm{t}}} \\ \mathrm{t}=\mathrm{time}\mathrm{to}\mathrm{traveled}\mathrm{the}\mathrm{car} \\ \mathrm{v}=\mathrm{velocity}\mathrm{of}\mathrm{car} \\ \mathrm{d}=\frac{{\mathrm{v}}^2}{2{\mathrm{a}}_{\mathrm{t}}} \\ {\mathrm{a}}_{\mathrm{r}}\mathrm{r}={\mathrm{v}}^2 \\ \mathrm{d}=\frac{{\mathrm{a}}_{\mathrm{r}}\mathrm{r}}{2{\mathrm{a}}_{\mathrm{t}}} \\ {\mathrm{a}}_{\mathrm{r}}=\mathrm{radical}\mathrm{acceleration} \end{gathered}$$

By putting values of distance and radius in angle formula:

localid="1648098776664" $$\begin{gathered} \mathrm{\theta }=\frac{{\mathrm{a}}_{\mathrm{r}}\mathrm{r}}{2{\mathrm{a}}_{\mathrm{t}}\mathrm{r}} \\ \mathrm{\theta }=\frac{{\mathrm{a}}_{\mathrm{r}}}{2{\mathrm{a}}_{\mathrm{t}}} \\ \mathrm{\theta }=\frac{1}{2}\sqrt{\frac{{\mathrm{a}}^2}{{{\mathrm{a}}_{\mathrm{t}}}^2}-1} \end{gathered}$$

Now putting the values in formula, the angle is:

$$\begin{gathered} \mathrm{\theta }=\frac{1}{2}\sqrt{\frac{2^2}{1^2}-1} \\ \mathrm{\theta }=0.866\mathrm{rad} \end{gathered}$$

Converting angle from radian to degree:

localid="1649071100890" $$\begin{gathered} \mathrm{\theta }=0.866\mathrm{rad}\times \frac{180°}{\mathrm{\pi }\mathrm{rad}} \\ \mathrm{\theta }=49.64° \end{gathered}$$