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

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

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

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) \(v = t^{2} − 10t + 21\)
    2. (b) \(v = t^{2} − 10t + 25\)
    3. (c) \(v = 6\cos(2t)\) (first time only)

  2. A tram moves along a straight street. Its velocity is \(v = t^{2} − 9t + 14\) m s⁻¹ at time \(t\) seconds.

    1. (a) Find its displacement from the start in the first 8 seconds.
    2. (b) Find the total distance it travels in the first 8 seconds.

  3. A pendulum bob moves along a straight line. Its velocity is \(v = 12\cos(0.4t) − 5\) m s⁻¹ for \(0 \le t \le 10\).

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

  4. A piston moves along a straight cylinder. Its displacement from the base of the cylinder is \(s = t^{3} − 15t^{2} + 63t\) cm at time \(t\) seconds. Decide whether it is speeding up or slowing down:

    1. (a) when \(t = 2\)
    2. (b) when \(t = 4\)
    3. (c) when \(t = 6\)

In context

  1. A lift moves up and down a straight shaft. Its velocity is \(v = 3t^{2} − 21t + 30\) m s⁻¹ at time \(t\) seconds, for \(0 \le t \le 6\), where positive means upwards.

    1. (a) Find the times when it is at rest.
    2. (b) Find its displacement and the total distance it travels in the 6 seconds.
    3. (c) Is it speeding up or slowing down when \(t = 4.5\)?

  2. A test car drives along a straight track. The graph shows its velocity \(v\) in m s⁻¹ at time \(t\) seconds, where positive means forwards.

    2468101214−6−4−22468101214t (s)v
    1. (a) Find the acceleration during the first 4 seconds.
    2. (b) Find the time at which the car changes direction.
    3. (c) Find its displacement from the start at the end of the 14 seconds.
    4. (d) Find the total distance it travels in the 14 seconds.

  3. A toy train runs along a straight track. The graph shows its displacement \(s\) metres from the station at time \(t\) seconds.

    12345678910111213481216202428323640t (s)s
    1. (a) Find the times when it is at rest.
    2. (b) Find its displacement from the start at \(t = 12\).
    3. (c) Find the total distance it travels in the 12 seconds.
    4. (d) Find the time when it is back at its starting position.

Stretch

  1. A cable car moves along a straight line. Its velocity is \(v = 9\sin(0.5t) − 3\) m s⁻¹ for \(0 \le t \le 10\). 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 10 seconds.

  2. A block slides on a straight rail, starting at the origin. Its velocity is \(v = 3t^{2} − 26t + 16\) 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