An object moves along a coordinate axis so that its position at time t is given by s(t)=t3. Estimate its instantaneous velocity at time t=2 by computing its average velocity over the time interval [2,2.001].

If the position at time t is

s(t) = (1 m/s³) t ³

then the average velocity over t = 2 s and t = 2.001 s is

v (ave) = (s (2.001 s) - s (2 s)) / (2.001 s - 2 s)

v (ave) = ((1 m/s³) (2.001 s)³ - (1 m/s³) (2 s)³) / (2.001 s - 2 s)

v (ave) ≈ (8.01201 m - 8 m) / (0.001 s)

v (ave) ≈ 12.006 m/s

468

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