A car moves in positive y-direction with velocity v proportional to distance travelled y as `v(y) prop y^(beta)`, where `beta` is a positive constant. The car covers a distance L with average velocity `lt v gt` proportional to L as `lt v gt prop L^(1//3)`, The constant `beta` is given as

Options

A. `(1)/(4)`
B. `(1)/(3)`
C. `(2)/(3)`
D. `(1)/(2)`

Answer

B

Explanation

Relation between average velocity and instantaneous velocity is
`lt v gt =(1)/(t_(2)-t_(1))int_(t_(1))^(t_(2))v(t)dt`
In the given problem, `v(t) prop y^(beta)` (independent of t)
So, `" "lt v lt prop v(t)` or `(v) prop y^(beta)`
`implies" "log v+v=-logx-logc`
`implies" "log((y)/(x))+((y)/(x))=-log(cx)`
`log((y)/(x)xx cx)=-(y)/(x)impliescy=e^(-(y)/(x))impliese^(-(y)/(x))=cy`