A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time t is proportional to :
Answer & explanation
Correct answer: option 4
Let us assume that the displacement of the body is directly proportional to \(t^n\), i.e. :
\(s = Kt^n\)
Thus, the velocity : \(v = \frac{ds}{dt} = Kn t^{n-1}\)
and the acceleration : \(a = \frac{dv}{dt} = Kn(n-1)t^{n-2}\)
The Force : \(F = ma = mKn(n-1)t^{n-2}\)
Hence, Power : \(P = Fv = (mKn(n-1)t^{n-2})(Kn t^{n-1})\)
\(P = mKn^2(n-1)t^{2n-3}\)
As power is constant, Thus, independent of time.
Hence : 2n - 3 = 0
\(\Rightarrow n = \frac{3}{2}\)
Thus, \(s \propto t^{3/2}\)