Question

82.
The velocity ‘v’ reachedby a car of mass ‘m’ on moving a certaindistance from the starting pont when driven by a motor with constant power ‘P’ is such that

Answer 1

The velocity v will be such that  $v =( \frac{3Px}{m} )^{\frac{1}{3} }$ .

Solution :

• The car is driven with motor which has constant power P.
• Power is given by ,    

        P = F.v    where F is the force.

• Now, F = m.a    where a is the acceleration.

• Also,    $a = \frac{dv}{dt}$  .
• Hence we get ,

        $P = (m \frac{dv}{dt} ) v$

     ∴ $\frac{P}{m} = v \frac{dv}{dt}$    ....(1)

• Now,   $\frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt}$  
• We know that,    $\frac{dx}{dt} = v$  
• Hence, equation (1) becomes,

        $v^{2} \frac{dv}{dx} = \frac{P}{m}$    

    ∴  $v^{2}dv = \frac{P}{m} dx$  

• Integrating both sides we get,

        $\frac{v^{3} }{3} = \frac{Px}{m}$  

      Hence,   $v = (\frac{3Px}{m})^{\frac{1}{3} }$