Motion in two dimensions with vectorsAQA A-Level Maths: Subtopic test

1. 1
   A particle $P$ moves in a horizontal plane with constant acceleration $(2\mathbf i-\mathbf j)$ m s$^{-2}$, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors. At time $t=0$ the velocity of $P$ is $(\mathbf i+3\mathbf j)$ m s$^{-1}$ and its position vector relative to a fixed origin $O$ is $(3\mathbf i+4\mathbf j)$ m.

   (a)

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

   [1 mark]

   • A$(7\mathbf i+6\mathbf j)$ m s$^{-1}$
   • B$7\mathbf i$ m s$^{-1}$
   • C$(3\mathbf i+2\mathbf j)$ m s$^{-1}$
   • D$(6\mathbf i-3\mathbf j)$ m s$^{-1}$

   (b)

   Find the position vector of $P$ when $t=3$.

   [1 mark]

   • A$(15\mathbf i+8.5\mathbf j)$ m
   • B$(24\mathbf i+4\mathbf j)$ m
   • C$(15\mathbf i+17.5\mathbf j)$ m
   • D$(12\mathbf i+4.5\mathbf j)$ m

   (c)

   Find the speed of $P$ when $t=2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves in a plane so that at time $t$ seconds its position vector relative to a fixed origin $O$ is $\mathbf r=\left[(t^3-3t)\mathbf i+(4t-t^2)\mathbf j\right]$ m, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors.

   (a)

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

   [1 mark]

   • A$(18\mathbf i+3\mathbf j)$ m s$^{-1}$
   • B$(24\mathbf i+2\mathbf j)$ m s$^{-1}$
   • C$(24\mathbf i-2\mathbf j)$ m s$^{-1}$
   • D$(24\mathbf i-6\mathbf j)$ m s$^{-1}$

   (b)

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

   [1 mark]

   • A$9\mathbf i$ m s$^{-2}$
   • B$(6\mathbf i-2\mathbf j)$ m s$^{-2}$
   • C$12\mathbf i$ m s$^{-2}$
   • D$(12\mathbf i-2\mathbf j)$ m s$^{-2}$

   (c)

   Find the value of $t$ at which the particle is moving parallel to $\mathbf i$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle moves in a plane with acceleration $\mathbf a=(4t\,\mathbf i-2\mathbf j)$ m s$^{-2}$ at time $t$ seconds, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors. When $t=0$ the velocity of the particle is $(2\mathbf i+5\mathbf j)$ m s$^{-1}$ and its position vector relative to a fixed origin $O$ is $(\mathbf i+3\mathbf j)$ m.

   (a)

   Find the velocity of the particle at time $t$.

   [3 marks]

   (b)

   Find the position vector of the particle when $t=3$, and find its distance from $O$ at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle moves in a horizontal plane. At time $t$ seconds, $t\ge0$, its velocity is $\mathbf v=\left[(3t^2-12t+9)\mathbf i+(4t-8)\mathbf j\right]$ m s$^{-1}$, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors. When $t=0$ the particle is at the point with position vector $(2\mathbf i-\mathbf j)$ m relative to a fixed origin $O$.

   (a)

   (i) Find the acceleration of the particle when $t=1$.
   (ii) Show that the particle is never at rest.
   (iii) Find the speed of the particle when $t=2$.

   [6 marks]

   (b)

   (i) Find the position vector of the particle at time $t$.
   (ii) Find the position vector of the particle at the instant when it is moving parallel to $\mathbf i$.

   [6 marks]

   Total for question 4: 12 marks

End of questions