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Physics · MOTION IN A STRAIGHT LINE
हिन्दीConsider the following table showing the calculation of limiting value of \( \frac{\Delta x}{\Delta t} \) centered at \( t = 4.0\text{ s} \) for position function \( x(t) = 0.08 t^3 \): | \( \Delta t\text{ (s)} \) | \( t_1\text{ (s)} \) | \( t_2\text{ (s)} \) | \( x(t_1)\text{ (m)} \) | \( x(t_2)\text{ (m)} \) | \( \Delta x\text{ (m)} \) | \( \frac{\Delta x}{\Delta t}\text{ (m s}^{-1}\text{)} \) | |---|---|---|---|---|---|---| | 2.0 | 3.0 | 5.0 | 2.16 | 10.0 | 7.84 | 3.92 | | 1.0 | 3.5 | 4.5 | 3.43 | 7.29 | 3.86 | 3.86 | | 0.5 | 3.75 | 4.25 | 4.21875 | 6.14125 | 1.9225 | 3.845 | What is the limiting value of the average velocity as \( \Delta t \to 0 \) (i.e., the instantaneous velocity at \( t = 4.0\text{ s} \))?
Correct answer · Option 2
\( 3.84\text{ m s}^{-1} \)
Explanation
As \( \Delta t \) decreases towards zero, \( \frac{\Delta x}{\Delta t} \) approaches \( \frac{dx}{dt} \). Differentiating \( x(t) = 0.08 t^3 \) gives \( v(t) = 0.24 t^2 \). At \( t = 4.0\text{ s} \), \( v(4) = 0.24 \times 16 = 3.84\text{ m s}^{-1} \).
