Displacement, velocity, acceleration and force as vectorsEdexcel International A Level Maths: Subtopic test

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

   (a)

   What is the speed of the particle?

   [1 mark]

   • A7 m s⁻¹
   • B5 m s⁻¹
   • C25 m s⁻¹
   • D1 m s⁻¹

   (b)

   What is the displacement of the particle in the first 6 s?

   [1 mark]

   • A$(20\mathbf{i}-19\mathbf{j})$ m
   • B$(0.5\mathbf{i}-0.67\mathbf{j})$ m
   • C$(3\mathbf{i}-4\mathbf{j})$ m
   • D$(18\mathbf{i}-24\mathbf{j})$ m

   (c)

   Find the position vector of the particle at $t=6$ s.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves in a plane with constant acceleration. At time $t=0$ its velocity is $(2\mathbf{i}+\mathbf{j})$ m s$^{-1}$ and 4 s later its velocity is $(14\mathbf{i}-8\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

   Which expression gives the acceleration of the particle?

   [1 mark]

   • A$\dfrac{14\mathbf{i}-8\mathbf{j}}{4}$
   • B$\dfrac{(14\mathbf{i}-8\mathbf{j})+(2\mathbf{i}+\mathbf{j})}{4}$
   • C$\dfrac{(14\mathbf{i}-8\mathbf{j})-(2\mathbf{i}+\mathbf{j})}{4}$
   • D$4\left[(14\mathbf{i}-8\mathbf{j})-(2\mathbf{i}+\mathbf{j})\right]$

   (b)

   What is the acceleration of the particle?

   [1 mark]

   • A$(3\mathbf{i}-2.25\mathbf{j})$ m s$^{-2}$
   • B$(12\mathbf{i}-9\mathbf{j})$ m s$^{-2}$
   • C$(4\mathbf{i}-1.75\mathbf{j})$ m s$^{-2}$
   • D$(-3\mathbf{i}+2.25\mathbf{j})$ m s$^{-2}$

   (c)

   Find the magnitude of the acceleration.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Relative to a fixed origin $O$, a ship $S$ moves with constant velocity $(4\mathbf{i}+3\mathbf{j})$ km h$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north. At time $t=0$ (hours) the ship is at the point with position vector $(-8\mathbf{i}-5\mathbf{j})$ km. A lighthouse $L$ is at the point with position vector $(12\mathbf{i}+4\mathbf{j})$ km.

   (a)

   Find the position vector of $S$ at time $t$ hours. Hence find the distance of $S$ from $O$ when $t=2$.

   [3 marks]

   (b)

   Find the time at which $S$ is due north of $L$, and find the distance $SL$ at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drone moves in a horizontal plane, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north. At time $t=0$ its velocity is $(3\mathbf{i}+4\mathbf{j})$ m s$^{-1}$ and it then moves with constant acceleration $(-0.5\mathbf{i}+0.75\mathbf{j})$ m s$^{-2}$.

   (a)

   Find the velocity of the drone at $t=4$ s. Hence find its speed and the angle its velocity makes with the direction of $\mathbf{i}$, to 1 decimal place.

   [6 marks]

   (b)

   Find the time at which the drone is moving due north, and its speed then. Find also the angle between its velocity and the direction of $-\mathbf{i}$ when $t=10$ s.

   [6 marks]

   Total for question 4: 12 marks

End of questions