More practice questions

4. Q4. A particle moves with r = (4 cos t)î + (4 sin t)ĵ m (t in s; the angle t is in radians). Show that its speed is constant and find the angle between v and a.

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   v = −4 sin t î + 4 cos t ĵ, so v = 4 m s^⁻¹ always. a = −4 cos t î − 4 sin t ĵ m s^⁻², numerically −r, so it points to the origin. v·a = 16 sin t cos t − 16 sin t cos t = 0, so the angle is 90°: the particle moves on a circle of radius 4 m at constant speed.

5. Q5. Between t = 0 and t = 2 s a particle’s velocity changes from 4î m s^⁻¹ to 4ĵ m s^⁻¹. Find its average acceleration.

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   Δv = −4î + 4ĵ, so the average acceleration is (−2î + 2ĵ) m s^⁻², magnitude 2√2 ≈ 2.83 m s^⁻², pointing north-west if î is east and ĵ north. The speed is 4 m s^⁻¹ at both ends, yet the average acceleration is not zero: |Δv| is not Δ|v|.

6. Q6. For r = t^³î − 2t^²ĵ m (t in s, t ≥ 0), is the velocity ever perpendicular to the acceleration?

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   v = 3t^²î − 4tĵ; a = 6tî − 4ĵ. v·a = 18t^³ + 16t = 2t(9t^² + 8), which is zero only at t = 0, where v = 0 and there is no angle. For t > 0 both factors are positive, so v·a > 0: the angle stays below 90° and the particle keeps speeding up. They are never perpendicular.