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Physics · MOTION IN A STRAIGHT LINE
हिन्दीReferring to the position-time graph shown in Fig. 2.1, the position of a particle is given by \( x(t) = 0.08 t^3 \) (where \( x \) is in meters and \( t \) is in seconds). Calculate the percentage error in approximating the true instantaneous velocity at \( t = 4.0 \text{ s} \) by taking the average velocity over the finite symmetric time interval \( \Delta t = 2.0 \text{ s} \) centered at \( t = 4.0 \text{ s} \).
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Correct answer · Option C
\( 2.08\% \)
Explanation
True instantaneous velocity \( v_{\text{inst}} = \left.\frac{dx}{dt}\right|_{t=4} = 3(0.08)(4^2) = 3.84 \text{ m/s} \). Interval \( [3, 5] \text{ s} \): \( x(5) = 10.0 \text{ m} \), \( x(3) = 2.16 \text{ m} \). Average velocity \( v_{\text{avg}} = \frac{10.0 - 2.16}{2} = 3.92 \text{ m/s} \). Percentage error = \( \frac{3.92 - 3.84}{3.84} \times 100\% = \frac{0.08}{3.84} \times 100\% \approx 2.08\% \).
