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IB Mathematics: Applications and Interpretation HL · AHL 4: Vectors · AHL 4.27

Name Date

A second object that starts later Problem set B

Replace \(t\) by \((t - a)\) for an object that starts \(a\) seconds later. Compare the two objects at the same clock time \(t\), then take the magnitude of the difference of the two displacements. Use \(g = 9.8\) m s⁻².

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. Two jets of water leave the same nozzle. The displacement of the first droplet from its launch point at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 6t \\ 13t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second droplet is launched from the same point along the same path, 0.7 seconds later. Time \(t\) is still measured from the launch of the first.

    1. (a) Write down the displacement of the second droplet at time \(t\), for \(t \ge 0.7\).
    2. (b) Write down the velocity of the second droplet at time \(t\).

  2. Two skaters follow the same marked path. The displacement of the first from its start at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 0.5t^2 \\ 4t \end{pmatrix}\) m. The second follows the same path from the same start, but begins 3 seconds later. Time \(t\) is measured from the start of the first.

    1. (a) Write down the displacement of the second at time \(t\), for \(t \ge 3\).
    2. (b) Find the position of the second when \(t = 6\).

  3. Two footballs are kicked from the same spot. The displacement of the first ball at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 6t \\ 11t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second ball follows the same path, launched 0.5 seconds later. Find the distance between the two balls 1.4 seconds after the second is launched.

In context

  1. Two balloons drift near a hill. Time \(t\) is in seconds from when the first balloon starts. The first balloon has position vector \(\mathbf{r}_P = \begin{pmatrix} -10 \\ 5 \\ 30 \end{pmatrix} + t\begin{pmatrix} 4 \\ -1 \\ -2 \end{pmatrix}\) m. The second balloon starts \(3\) seconds later and has position vector \(\mathbf{r}_Q = \begin{pmatrix} 40 \\ -20 \\ 10 \end{pmatrix} + (t - 3)\begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}\) m for \(t \ge 3\). Find the distance between them when \(t = 8\).

  2. Two rockets are launched from the same stand. The displacement of the first rocket at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 4t \\ 13t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second rocket follows the same path, launched 1 second after the first.

    1. (a) Write down the displacement of the second rocket at time \(t\).
    2. (b) Find the distance between the two rockets 1.6 seconds after the second is launched.

Stretch

  1. Two javelins are thrown from the same mark. The displacement of the first javelin at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 7t \\ 11t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second javelin follows the same path, launched 0.6 seconds later.

    1. (a) Find the time at which the two javelins are at the same height.
    2. (b) Find the distance between the two javelins at that time.

jmaths.xyz · AI HL · AHL 4.27 An object that starts later