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In the figure the variation of components of acceleration of a particle of mass 1 kg is shown with respect to time. The initial velocity of the particle is $\vec{u} = (- 3 \hat{i} + 4 \hat{j}) m / s .$ The total work done by the resultant force on the particle in time interval from $t = 0 t o t = 4$ second is

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None of these

22.5 J

10 J

answer is B.

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Detailed Solution

From the given graphs :

$a_{x} = \frac{3}{4} t and a_{y} = - (\frac{3}{4} t + 1) \Rightarrow v_{x} = \frac{3}{8} t^{2} + C .$

$At t = 0, v_{x} = - 3 \Rightarrow C = - 3 .$

$∴ v_{x} = \frac{3}{8} t^{2} - 3 \Rightarrow d x = (\frac{3}{8} t^{2} - 3) d t . . . . . . (i)$

Similarly, $d y = (- \frac{3}{2} t^{2} - t + 4) d t . . . . . . (i i)$

$As d W = \vec{F} \cdot d \vec{s} = \vec{F} \cdot (d x \hat{i} + d y \hat{j})$

$∴ \int_{0}^{w} d W = \int_{0}^{w} [\frac{3}{4} t \hat{i} - (\frac{3}{4} t + 1) \hat{j}] \cdot [(\frac{3}{8} t^{2} - 3) \hat{i} + (\frac{3}{8} t^{2} - t + 4) \hat{j}] d t$

$∴ W = 10 J$

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