Question 301: Work, Energy and Power & Circular Motion

Question

A body of mass 2 kg is driven by an engine
delivering a constant power of 1 J/s. The body
starts from rest and moves in a straight line.
After 9 seconds, the body has moved a
distance (in m) _______.

Accepted Answer

Let s be the required distance.
From Work - Energy theorem,
Work = Change in kinetic energy
$$\Rightarrow$$ Power $$	imes$$ Time = $$\Delta$$K
i.e., Pt = $$\Delta$$K $$\Rightarrow$$ Pt = $${1 \over 2}$$mv
2
..... (i)
Given, P = 1 Js
$$-$$1
, t = 9 s, m = 2 kg
Substituting all the given values in eq. (i), we get
1 $$	imes$$ 9 = $${1 \over 2}$$(2) v
2
v
2
= 9 $$\Rightarrow$$ v = 3 m/s (at t = 9 s)
As, Fv = P $$\Rightarrow$$ (ma)v = P [$$\because$$ F = ma]
$$ \Rightarrow m\left[ {{{dv} \over {dt}}} ight]v = P \Rightarrow m\left[ {{{ds} \over {dt}}{{dv} \over {ds}}} ight]v = P$$
$$ \Rightarrow m\left[ {v{{dv} \over {ds}}} ight]v = P$$
$$ \Rightarrow 2{v^2}dv = ds$$ {$$\because$$ P = 1 J/s and m = 2 kg}
Integrating both sides,
$$\int\limits_0^3 {2{v^2}dv = \int\limits_0^s {ds \Rightarrow {2 \over 3}[{v^3}]_0^3 = 8} } $$
$${2 \over 3}[27 - 0] = s \Rightarrow s = 18$$ m
Hence, after 9 s, the body has moved a distance of 18 m.

Work, Energy and Power & Circular Motion

A body of mass 2 kg is driven by an engine delivering a constant power of 1 J/s. The body starts from rest and moves in a straight line. After 9 seconds, the body has moved a distance (in m) _______.

Integer / Numeric Answer Type