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IB Mathematics: Applications and Interpretation HL · AHL 4: Vectors · AHL 4.22

Name Date

Motion on a line: direction changes, distance and speed Problem set A

Show your working. Use your calculator for integrals and for solving, in radian mode when sin or cos is involved. Velocities are in m s⁻¹ and times in seconds.

Difficulty: first skills applying them stretch

Skills

  1. Each velocity is for an object moving on a straight line, in m s⁻¹, with time \(t \ge 0\) in seconds. Find the time or times when the object is at rest, and say whether it changes direction at each.

    1. (a) \(v = t^{2} − 8t + 12\)
    2. (b) \(v = t^{2} − 6t + 9\)
    3. (c) \(v = 4\cos(0.5t)\) (first time only)

  2. A toy car moves along a straight track. Its velocity is \(v = t^{2} − 6t + 5\) m s⁻¹ at time \(t\) seconds.

    1. (a) Find its displacement from the start in the first 6 seconds.
    2. (b) Find the total distance it travels in the first 6 seconds.

  3. A model aircraft slides along a straight wire. Its velocity is \(v = 8\sin(0.5t) − 3\) m s⁻¹ for \(0 \le t \le 8\).

    1. (a) Find the greatest velocity.
    2. (b) Find the least velocity.
    3. (c) Find the greatest speed.

  4. A bead moves along a straight wire. Its displacement from the centre of the wire is \(s = 2t^{3} − 21t^{2} + 60t\) cm at time \(t\) seconds. Decide whether it is speeding up or slowing down:

    1. (a) when \(t = 1\)
    2. (b) when \(t = 3\)
    3. (c) when \(t = 4\)

In context

  1. A trolley runs along a straight laboratory track. Its velocity is \(v = 2t^{2} − 14t + 24\) m s⁻¹ at time \(t\) seconds, for \(0 \le t \le 5\), where positive means towards the end stop.

    1. (a) Find the times when it is at rest.
    2. (b) Find its displacement and the total distance it travels in the 5 seconds.
    3. (c) Is it speeding up or slowing down when \(t = 3.2\)?

  2. A remote-controlled boat moves along a straight canal. The graph shows its velocity \(v\) in m s⁻¹ at time \(t\) seconds, where positive means towards the lock.

    1234567891011121314−10−8−6−4−2246810t (s)v
    1. (a) Find the acceleration during the first 2 seconds.
    2. (b) Find the time at which the boat changes direction.
    3. (c) Find its displacement from the start at the end of the 13 seconds.
    4. (d) Find the total distance it travels in the 13 seconds.

  3. A robot moves along a straight rail. The graph shows its displacement \(s\) metres from the charging point at time \(t\) seconds.

    1234567892468101214161820t (s)s
    1. (a) Find the times when it is at rest.
    2. (b) Find its displacement from the start at \(t = 8\).
    3. (c) Find the total distance it travels in the 8 seconds.
    4. (d) Find the time when it is back at its starting position.

Stretch

  1. A float on a river moves along a straight line. Its velocity is \(v = 6\sin(0.8t) − 2\) m s⁻¹ for \(0 \le t \le 8\). The calculator must be in radians.

    1. (a) Find the first time it is at rest.
    2. (b) Find the greatest speed in the interval.
    3. (c) Find the total distance it travels in the 8 seconds.

  2. A particle moves on a straight line, starting at the origin. Its velocity is \(v = 3t^{2} − 20t + 12\) m s⁻¹ at time \(t\) seconds.

    1. (a) Find an expression for its displacement from the start at time \(t\).
    2. (b) Find the first time after the start when it is back at its starting position.
    3. (c) Find its velocity at that time. In which direction is it moving?

jmaths.xyz · AI HL · AHL 4.22 Motion on a line: direction changes, distance and speed