Q. Show that the speed reached by a car of mass `m` that is driven with constant power P is given by:- `v= ((4xP)/m)^(1/3)` where `x` is the distance travelled starting from rest.

Expert Answers
justaguide eNotes educator| Certified Educator

A constant power P drives a car of mass m. If the acceleration due to the applied power is a and the car travels a distance x in time t, `P = (m*a*x)/t`

The velocity of the car after t seconds with an acceleration a is given by `v^2 - 0^2 = 2*a*x`

Also, `v = 0 + a*t`

=> `t = v/a`

`P = (m*a*x)/t`

= `(m*a*x)/(v/a)`

= `(m*a^2*x)/v`

=> `v = (m*a^2*x)/P`

`v^2 = 2*a*x`

=> `a = (v^2)/(2*x)`

`v = (m*((v^2)/(2*x))^2*x)/P`

=> `v = (m*((v^2)/(2*x))^2*x)/P`

=> `v = (m*v^4*x)/(4*x^2*P)`

=> `v^3 = (4*x^2*P)/(m*x)`

=> `v^3 = (4*x*P)/m`

=> `v = ((4*x*P)/m)^(1/3)`

The speed of the car after traveling a distance x is `v = ((4*x*P)/m)^(1/3)`