Question

the engine of a car is producing constant power . the car starts from rest and continues to move along a straight line. at time t , let x be the displacement , V be the velocity and 'a' be the acceleration of the car then​

Answer 1

Given:

The engine of a car is producing constant power . the car starts from rest and continues to move along a straight line. at time t , let x be the displacement , v be the velocity and 'a' be the acceleration of the car.

To find:

Displacement function ?

Solution:

Power can be written as the force multiplied with velocity.

$\therefore \: P = force \times v$

$\implies\: P = (ma) \times v$

$\implies\: P = m \times \dfrac{dv}{dt} \times v$

$\implies\: P = m \times (\dfrac{dv}{dx} \times \dfrac{dx}{dt} ) \times v$

$\implies\: P = m \times (\dfrac{dv}{dx} \times v ) \times v$

$\implies\: P = m \times \dfrac{dv}{dx} \times {v}^{2}$

$\implies\: dx= \dfrac{m}{P} \: {v}^{2} \: dv$

$\implies\: \displaystyle \int_{0}^{x} dx= \dfrac{m}{P} \: \int_{0}^{v} {v}^{2} \: dv$

$\implies\: x= \dfrac{m {v}^{3} }{P}$

So, displacement function is :

$\boxed{ \bf\: x= \dfrac{m {v}^{3} }{P} }$