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Physics · LAWS OF MOTION
हिन्दीA body of mass \( m = 3\text{ kg} \) moves under the influence of a time-dependent force such that its position vector is given by \( \mathbf{r}(t) = (t^3 \hat{i} + 2t^2 \hat{j} - t \hat{k})\text{ m} \). Determine the magnitude of the net force acting on the body at \( t = 1\text{ s} \) and the magnitude of the impulse delivered to the body during the time interval from \( t = 0\text{ s} \) to \( t = 1\text{ s} \).
Correct answer · Option 3
\( |\mathbf{F}| = 6\sqrt{13}\text{ N} \) and \( |\mathbf{J}| = 15\text{ N s} \)
Explanation
\( \mathbf{v}(t) = \frac{d\mathbf{r}}{dt} = 3t^2 \hat{i} + 4t \hat{j} - \hat{k} \). At \( t = 1\text{ s} \), \( \mathbf{a}(1) = 6 \hat{i} + 4 \hat{j}\text{ m/s}^2 \implies \mathbf{F} = 3(6 \hat{i} + 4 \hat{j}) = 18 \hat{i} + 12 \hat{j}\text{ N} \implies |\mathbf{F}| = \sqrt{18^2 + 12^2} = 6\sqrt{13}\text{ N} \). Impulse \( \mathbf{J} = m(\mathbf{v}(1) - \mathbf{v}(0)) = 3(3 \hat{i} + 4 \hat{j}) = 9 \hat{i} + 12 \hat{j}\text{ N s} \implies |\mathbf{J}| = \sqrt{9^2 + 12^2} = 15\text{ N s} \).
