More practice questions

4. Q4. A particle moves with x = t^³ − 6t^² + 9t (metres, seconds). At what times is it momentarily at rest, and where is it then?

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   v = 3t^² − 12t + 9 = 3(t − 1)(t − 3), which is zero at t = 1 s and t = 3 s. Then x(1) = 1 − 6 + 9 = 4 m and x(3) = 27 − 54 + 27 = 0 m, back at the origin.

5. Q5. A car’s position is measured as 10.0 m at t = 2.0 s and 17.6 m at t = 3.0 s. Estimate its velocity at t = 2.5 s. When is this estimate exact?

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   Using the interval centred on 2.5 s: (17.6 − 10.0)/(3.0 − 2.0) = 7.6 m/s. The estimate is exact if the acceleration is constant over the interval (then the x-t graph is a parabola, and the chord is parallel to the tangent at the mid-time); otherwise it improves as the interval shrinks.

6. Q6. For x = 2 + 5t − t^² (metres, seconds), at what times is the speed 3 m/s?

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   v = 5 − 2t. Speed 3 m/s means v = +3 or v = −3, so t = 1 s or t = 4 s. At 1 s it moves in the positive direction; at 4 s it is coming back.