Question

Question: -A certain light truck can go around a curve having a radius of with a maximum speed of . To have the same acceleration, at what maximum speed can it go around a curve having a radius of ?

(a) 64 m/s

(b) 45 m/s

(c) 32 m/s

(d) 23 m/s

(e) 16 m/s

Short answer

The correct option is23m/s

Step-by-step solution

The acceleration between the tyres and the level road

The requisite centripetal acceleration is provided by friction between the tyres and the level road, allowing the vehicle to handle the curve safely without skidding.

${\mathbf{a}}_{\mathbf{c}\mathbf{,}\mathbf{max}}\mathbf{=}\frac{{\mathbf{v}}_{{\mathbf{max}}^{\mathbf{2}}}}{\mathbf{R}}$

After insertion of the values,

$$\begin{gathered} {\mathbf{a}}_{\mathbf{c}\mathbf{,}\mathbf{max}}\mathbf{}\mathbf{=}\mathbf{}\frac{{\mathbf{(}\mathbf{32}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}}^{\mathbf{2}}}{\mathbf{150}\mathbf{}\mathbf{m}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{82}\overline{\mathbf{6}}\mathbf{}{\mathbf{ms}}^{\mathbf{-}\mathbf{2}} \end{gathered}$$

The maximum safe speed 

The maximum safe speed from (1) becomes
$$\begin{gathered} v_{\text{new,}\max }\mathit{}\mathit{=}\mathit{}\sqrt{{\mathit{a}}_{\mathit{c}\mathit{,}\mathit{max}}{\mathit{R}}_{\mathit{\text{new}}}} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\sqrt{\mathit{6}\mathit{.}\mathit{82}\overline{\mathit{6}}\mathit{}\mathit{m}\mathit{/}{\mathit{s}}^{\mathit{2}}\mathit{}\mathit{x}\mathit{}\mathit{75}\mathit{.}\mathit{0}\mathit{}\mathit{m}} \\ \mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{}\mathit{=}\mathit{22}\mathit{.}\mathit{6}\mathit{}m{s}^{\mathit{-}\mathit{1}} \end{gathered}$$

(rounded to one decimal place)

So, the correct answer is 23 m/s.