Question

If acceleration of particle a = 2x, where x is position then velocity as a function of
position is given by

Answer 1

Given :   acceleration of particle a = 2x, where x is position  

To Find : velocity as a function of position is given by

Solution:

a  = 2x

a =  v (dv/dx)

=>   v dv/dx  = 2x

=> v dv  = 2x dx

integrating both sides

=>   v²/2  =  x²   + c            c is constant

=>  v² = 2 x²   + C              C  is constant

=>   v  = √(2 x²   + C)

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Answer 2

Given:

Acceleration of particle a = 2x, where x is position

To find:

Velocity as a function of position.

Calculation:

$\therefore \: a = 2x$

$\implies \: \dfrac{dv}{dt} = 2x$

• Applying Chain Rule:

$\implies \: \dfrac{dv}{dx} \times \dfrac{dx}{dt} = 2x$

$\implies \: \dfrac{dv}{dx} \times v = 2x$

$\implies \: v \dfrac{dv}{dx} = 2x$

$\implies \: v \: dv= 2x \: dx$

• Integrating on both sides:

$\displaystyle \implies \: \int v \: dv= \int2x \: dx$

$\displaystyle \implies \: \dfrac{ {v}^{2} }{2} = {x}^{2} + c$

• 'c' is constant.

$\displaystyle \implies \: {v}^{2} = 2{x}^{2} + k$

• let k = 2c.

$\displaystyle \implies \: v = \sqrt{ 2{x}^{2} + k}$

So, final answer is :

v = √(2x² + k)