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Question

A 10kg body moves along X-axis from $x = 0$ to $x = 5 m$ under the influence of a force given by $F = 5 - 2 x + 3 x^{2}$ . If the body starts from rest, the velocity at $x = 5 m$ is

Moderate

A

$10 m s^{- 1}$
B

$5 m s^{- 1}$
C

$9 m s^{- 1}$
D

$15 m s^{- 1}$

Solution

$\begin{array}{l} W = {KE}_{f} - {KE}_{i} \\ \int_{0}^{5} F d x = \frac{1}{2} m (v^{2} - 0^{2}) \\ \int_{0}^{5} (5 - 2 x + 3 x^{2}) d x = \frac{1}{2} \times 10 (v^{2} - 0^{2}) \\ 5 [x]_{0}^{5} - {[\frac{2 x^{2}}{2}]}_{0}^{5} + {[\frac{3 x^{3}}{3}]}_{0}^{5} = \frac{1}{2} \times 10 \times v^{2} \\ 25 - 25 + 125 = \frac{1}{2} \times 10 \times v^{2} \\ 125 = 5 v^{2} \\ v = 5 {ms}^{- 1} \end{array}$

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