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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 A

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.

Difficulty: first skills applying them stretch

Skills

  1. Two balls are thrown from the same spot. The displacement of the first ball from its launch point at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 7t \\ 9t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second ball is launched from the same point along the same path, 0.4 seconds later. Time \(t\) is still measured from the launch of the first.

    1. (a) Write down the displacement of the second ball at time \(t\), for \(t \ge 0.4\).
    2. (b) Write down the velocity of the second ball at time \(t\).

  2. Two toy trains run along the same track. The displacement of the first from its start at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} t^2 \\ 3t \end{pmatrix}\) m. The second follows the same path from the same start, but begins 2 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 \(t \ge 2\).
    2. (b) Find the position of the second when \(t = 5\).

  3. Two stones are thrown from a bridge. The displacement of the first stone at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 8t \\ 14t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second stone follows the same path, launched 0.7 seconds later. Find the distance between the two stones 1.3 seconds after the second is launched.

In context

  1. Two drones fly in the same airspace. Time \(t\) is in seconds from when the first drone starts. The first drone has position vector \(\mathbf{r}_P = \begin{pmatrix} 0 \\ 0 \\ 20 \end{pmatrix} + t\begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}\) m. The second drone starts \(4\) seconds later and has position vector \(\mathbf{r}_Q = \begin{pmatrix} 60 \\ -10 \\ 5 \end{pmatrix} + (t - 4)\begin{pmatrix} -3 \\ 4 \\ 1 \end{pmatrix}\) m for \(t \ge 4\). Find the distance between them when \(t = 10\).

  2. Two fireworks are fired from the same launcher. The displacement of the first firework at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 3t \\ 16t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second firework follows the same path, launched 1.5 seconds after the first.

    1. (a) Write down the displacement of the second firework at time \(t\).
    2. (b) Find the distance between the two fireworks 1.2 seconds after the second is launched.

Stretch

  1. Two arrows are shot from the same point. The displacement of the first arrow at time \(t\) seconds is \(\mathbf{r}_1 = \begin{pmatrix} 6t \\ 15t - 4.9t^2 \end{pmatrix}\) m, where the first component is horizontal and the second is upward. A second arrow follows the same path, launched 0.8 seconds later.

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

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