Question

A car moves in positive $y$ - direction with velocity $v$ proportional to distance travelled $y$ as $v(y) \propto y^{\beta}$, where $\beta$ is a positive constant. The car covers a distance $L$ with average velocity $< v >$ proportional to $L$ as $\left\langle v>\propto L^{1 / 3}\right.$ The constant $\beta$ is given as

• A. $\frac{1}{4}$
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

Relation between average velocity and instantaneous velocity is
$< v > =\frac{1}{t_{2}-t_{1}} \int_{t_{1}}^{t_{2}} v(t) d t$
In the given problem, $v(t) \propto y^{\beta}$ (independent of $t$ )
So, $\langle v\rangle \propto v(t) \text { or }(v) \propto y^{\beta}$
But according to question,
$< v >\propto y^{1 / 3}$
So, $\beta=\frac{1}{3}$