3.12 Vector applications to kinematicsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   A drone starts at the point with position vector $\mathbf{r}_0=\begin{pmatrix} 2 \\ 5 \end{pmatrix}$ m and moves with constant velocity $\mathbf{v}=\begin{pmatrix} 3 \\ -4 \end{pmatrix}$ m s$^{-1}$. Its position at time $t$ seconds is $\mathbf{r}=\mathbf{r}_0+t\mathbf{v}$.

   (a)

   Find the speed of the drone.

   [1 mark]

   • A$7$ m s$^{-1}$
   • B$1$ m s$^{-1}$
   • C$5$ m s$^{-1}$
   • D$\sqrt7$ m s$^{-1}$

   (b)

   Find the position vector of the drone when $t=4$.

   [1 mark]

   • A$\begin{pmatrix} 14 \\ 21 \end{pmatrix}$
   • B$\begin{pmatrix} 14 \\ -11 \end{pmatrix}$
   • C$\begin{pmatrix} 12 \\ -16 \end{pmatrix}$
   • D$\begin{pmatrix} 5 \\ 1 \end{pmatrix}$

   (c)

   Find the time at which the drone crosses the $x$-axis, and its $x$-coordinate at that time.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two boats, $A$ and $B$, move with constant velocities. Relative to a harbour at the origin (distances in km), the positions at time $t$ hours after noon are $\mathbf{r}_A=\begin{pmatrix} 1 \\ 2 \end{pmatrix}+t\begin{pmatrix} 4 \\ 1 \end{pmatrix}$ and $\mathbf{r}_B=\begin{pmatrix} 9 \\ -1 \end{pmatrix}+t\begin{pmatrix} -2 \\ 4 \end{pmatrix}$.

   (a)

   Find the position vector of $B$ relative to $A$ at noon ($t=0$), that is $\overrightarrow{AB}$.

   [1 mark]

   • A$\begin{pmatrix} 8 \\ -3 \end{pmatrix}$
   • B$\begin{pmatrix} -8 \\ 3 \end{pmatrix}$
   • C$\begin{pmatrix} 10 \\ 1 \end{pmatrix}$
   • D$\begin{pmatrix} 8 \\ 3 \end{pmatrix}$

   (b)

   Find the velocity of $B$ relative to $A$.

   [1 mark]

   • A$\begin{pmatrix} 6 \\ -3 \end{pmatrix}$
   • B$\begin{pmatrix} 2 \\ 5 \end{pmatrix}$
   • C$\begin{pmatrix} -2 \\ 4 \end{pmatrix}$
   • D$\begin{pmatrix} -6 \\ 3 \end{pmatrix}$

   (c)

   Use your GDC to find the times when the boats are exactly 5 km apart.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two drones, $D_1$ and $D_2$, fly in a large hall. With the origin at one corner of the floor (distances in metres), their positions $t$ seconds after launch are $\mathbf{r}_1=\begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}+t\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ and $\mathbf{r}_2=\begin{pmatrix} -1 \\ 9 \\ -1 \end{pmatrix}+t\begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix}$.

   (a)

   Show that the paths of the two drones intersect, and find the coordinates of the point of intersection.

   [3 marks]

   (b)

   Hence determine whether the drones collide. Find the distance between them at the instant $D_1$ is at the point of intersection.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball is kicked from the origin $O$ on level ground. With $x$ horizontal and $y$ vertically upwards (both in metres), its velocity at time $t$ seconds is $\mathbf{v}=\begin{pmatrix} 7 \\ 6-4t \end{pmatrix}$ m s$^{-1}$, for $t\geq0$.

   (a)

   (i) Write down the acceleration of the ball.
   (ii) Find the position vector of the ball at time $t$.
   (iii) Find the time at which the ball lands, and the horizontal distance it has travelled by then.

   [6 marks]

   (b)

   A second ball is kicked from $O$ at $t=2$ and moves exactly like the first ball but 2 seconds later: its position at time $t\geq2$ is $\mathbf{r}(t-2)$, where $\mathbf{r}(t)$ is the position of the first ball.
   (i) Write down the $y$-coordinate of the second ball at time $t$.
   (ii) Find the time at which the two balls are at the same height.
   (iii) Find the horizontal distance between the balls at that time.
   (iv) Determine whether the first ball is rising or falling at that time, justifying your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions