Question

A body of mass m starting from rest from origin moves along x axis with constant power p .Calculate relation between velocity /distance?

Answer 1

hope u get the solution

Answer 2

Answer:

The relation between velocity  and distance is  x ∝ $v^{3}$  .

Explanation:

Given , a body of mass = m

And the body is moving along x- axis with constant power = p

We know that  formula of work done  = F× d

Where F stand for force and d is the distance travel by the body .

Now ,  as we know that velocity is equal to rate of change of displacement .

⇒ v = $\frac{displacement }{Time taken } = \frac{d}{t}$

And power is equal to rate of doing work .

Power (p) = $\frac{work \ done }{Time \ time }$ = $\frac{w}{t}$

⇒p =  $\frac{F. d}{t}$ =  F v            [where w=F × d and v = $\frac{d}{t}$ ]

⇒p = mav                    [Force (F) = mass× acceleration]

⇒p = m ×$v \frac{dv}{dx}$ ×v            [acceleration 'a' =v$\frac{dv}{dx}$ ]

⇒ p = m$v^{2} \frac{dv}{dx}$

⇒pdx = m$v^{2}$ dv

Now integrated both side we get ,

p $\int^x_0$dx = m $\int ^v_0$$v^{2}$ dv

⇒px = m × $\frac{v^{3} }{3}$

 Hence, the relation between velocity and distance is  x ∝ $v^{3}$ or v ∝ $x^{\frac{1}{3} }$ .

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