Kinematics with vectors in two dimensionsEdexcel A-Level Maths: Subtopic test

1. 1
   A particle $P$ moves in a horizontal plane with constant acceleration $(4\mathbf{i}+2\mathbf{j})$ m s$^{-2}$. At time $t=0$ its velocity is $(2\mathbf{i}-3\mathbf{j})$ m s$^{-1}$.

   (a)

   Find the velocity of $P$ when $t=3$.

   [1 mark]

   • A$(12\mathbf{i}+6\mathbf{j})$ m s$^{-1}$
   • B$(14\mathbf{i}+3\mathbf{j})$ m s$^{-1}$
   • C$(14\mathbf{i}-9\mathbf{j})$ m s$^{-1}$
   • D$(6\mathbf{i}-\mathbf{j})$ m s$^{-1}$

   (b)

   Find the displacement of $P$ from its position at $t=0$ to its position at $t=2$.

   [1 mark]

   • A$(20\mathbf{i}+2\mathbf{j})$ m
   • B$(10\mathbf{i}+\mathbf{j})$ m
   • C$(12\mathbf{i}-2\mathbf{j})$ m
   • D$(4\mathbf{i}-6\mathbf{j})$ m

   (c)

   Find the value of $t$ at which $P$ is moving in the direction of $\mathbf{i}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle $Q$ moves in a plane. Its position vector relative to a fixed origin $O$ at time $t$ seconds is $\mathbf{r}=(t^3-6t)\mathbf{i}+(4t^2-3t)\mathbf{j}$ metres, for $t\ge0$.

   (a)

   Find the velocity of $Q$ when $t=2$.

   [1 mark]

   • A$(-4\mathbf{i}+10\mathbf{j})$ m s$^{-1}$
   • B$(12\mathbf{i}+8\mathbf{j})$ m s$^{-1}$
   • C$(6\mathbf{i}+5\mathbf{j})$ m s$^{-1}$
   • D$(6\mathbf{i}+13\mathbf{j})$ m s$^{-1}$

   (b)

   Find the acceleration of $Q$ when $t=2$.

   [1 mark]

   • A$(12\mathbf{i}+8\mathbf{j})$ m s$^{-2}$
   • B$(6\mathbf{i}+8\mathbf{j})$ m s$^{-2}$
   • C$(6\mathbf{i}+13\mathbf{j})$ m s$^{-2}$
   • D$(12\mathbf{i}+13\mathbf{j})$ m s$^{-2}$

   (c)

   Find the value of $t$ at which $Q$ is moving in the direction of $\mathbf{j}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   At time $t=0$ a particle is at the point $A$ with position vector $(3\mathbf{i}-2\mathbf{j})$ m, moving with velocity $(5\mathbf{i}+2\mathbf{j})$ m s$^{-1}$. The particle moves with constant acceleration. At time $t=4$ s it is at the point $B$ with position vector $(35\mathbf{i}+14\mathbf{j})$ m.

   (a)

   Find the acceleration of the particle.

   [3 marks]

   (b)

   Find the speed of the particle at $t=4$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A model boat moves on a lake. Relative to a fixed origin $O$, its position vector at time $t$ seconds is $\mathbf{r}=(t^3-6t^2+9t)\mathbf{i}+(2t^2-8t)\mathbf{j}$ metres, for $t\ge0$, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

   (i) Find an expression for the velocity of the boat at time $t$.
   (ii) Show that when $t=1$ the boat is moving parallel to $\mathbf{j}$, and state its speed.
   (iii) Find the acceleration of the boat when $t=1$.

   [6 marks]

   (b)

   The boat is at $O$ when $t=0$. Find the first time $t>0$ at which the boat is again on the line through $O$ parallel to $\mathbf{i}$, and find the distance of the boat from $O$ and its speed at that time.

   [6 marks]

   Total for question 4: 12 marks

End of questions