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Question

A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to


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Solution

Step 1: Given data

A body is moving unidirectionally under the influence of a source of constant power.

Step 2: Formula used

P=Wt[whereP=power,W=workdone,t=time]

Step 3: Calculating displacement

Now, the total work done by this force is equal to the product of the magnitude of applied force and the distance travelled by the body.

W=Fd

Substituting the value of work done in the power formula

P=Fdt

Now as the velocity is given by v=dt[d=distance,t=time] and the force is given by F=ma[m=mass,a=acceleration]

The power can be expressed as P=mav

Finally, we substitute v=at to obtain P=ma2t

a=Pmt......(i)

Now Newton’s first equation of motion gives the displacement of the body as d=ut+12at2............(ii)

We assume that the body was initially at rest, i.e., u=0.
So equation (ii) becomes d=12at2.......(iii)

Substituting equation (i) and (iii) we get, d=12Pmtt2

On simplifying we get,

d=12Pm×t2-12d=12Pm×t3/2dt3/2

Hence, its displacement is directly proportional to time t3/2.


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