First slide

Work

Question

The velocity of a 3kg body varies with time ‘t’ as $V= (2+ \frac{1}{2} t) {ms}^{-1}$ where ‘t’ is in seconds. The work done by the force during the time t=0s to t=2s______

Moderate

Solution

$\begin{array}{l} V = 2 + \frac{t}{2}; a = \frac{dv}{dt} = 0 + \frac{1}{2} = \frac{1}{2} {ms}^{- 1} \\ ds = Vdt = (2 + \frac{t}{2}) dt \\ W = \int \vec{F} \cdot \vec{ds} = \int mads \\ = \begin{aligned} \int_{0}^{2} 3 \times \frac{1}{2} (2 + \frac{t}{2}) dt \\ = \frac{3}{2} [2 \int_{0}^{2} dt + \frac{1}{2} \int_{0}^{2} tdt \end{aligned} \end{array}$
$\begin{array}{l} W = \frac{3}{2} [2 \times [t]_{0}^{2} + \frac{1}{2} {[\frac{t^{2}}{2}]}_{0}^{2}] \\ = \frac{3}{2} [2 \times 2 + \frac{1}{4} \times 4] \\ = \frac{3}{2} [5] = \frac{15}{2} = 7.5 J \end{array}$

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