Calculus in kinematicsAQA A-Level Maths: Subtopic test

1. 1
   A particle moves in a straight line. Its displacement from a fixed point $O$ at time $t$ seconds is $r=2t^3-9t^2+12t$ metres.

   (a)

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

   [1 mark]

   • A$9$ m s⁻¹
   • B$12$ m s⁻¹
   • C$18$ m s⁻¹
   • D$48$ m s⁻¹

   (b)

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

   [1 mark]

   • A$12$ m s⁻²
   • B$36$ m s⁻²
   • C$18$ m s⁻²
   • D$-6$ m s⁻²

   (c)

   Find the values of $t$ at which the particle is instantaneously at rest.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves in a straight line. At time $t$ seconds its acceleration is $(6t-4)$ m s⁻². When $t=0$ the particle is at the fixed point $O$ and has velocity $5$ m s⁻¹.

   (a)

   Which expression gives the velocity $v$ of the particle in m s⁻¹ at time $t$?

   [1 mark]

   • A$v=3t^2-4t+5$
   • B$v=3t^2-4t$
   • C$v=6t^2-4t+5$
   • D$v=t^3-2t^2+5t$

   (b)

   Find the displacement of the particle from $O$ when $t=2$.

   [1 mark]

   • A$9$ m
   • B$5$ m
   • C$8$ m
   • D$10$ m

   (c)

   Show that the particle is never instantaneously at rest.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves on a straight line through a fixed point $O$. At time $t$ seconds, $t\ge0$, its velocity is $v=3t^2-12t+9$ m s⁻¹. When $t=0$, $P$ is at $O$.

   (a)

   Find the acceleration of $P$ at each time when it is instantaneously at rest.

   [3 marks]

   (b)

   Find the total distance travelled by $P$ in the first $3$ seconds.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A remote-controlled car $P$ moves along a straight track. It starts at the point $O$ on the track when $t=0$, and for $0\le t\le6$ its velocity is $v=6t-t^2$ m s⁻¹, where $t$ is in seconds.

   (a)

   (i) Show that the speed of the car is greatest when $t=3$, and find this greatest speed.
   (ii) Find the displacement of the car from $O$ when $t=3$.

   [6 marks]

   (b)

   A second car $Q$ starts from $O$ at $t=0$ and travels along the track in the same direction at a constant speed of $4.5$ m s⁻¹. Find the time, other than $t=0$, at which the two cars are level, and give a reason for rejecting any other solution.

   [6 marks]

   Total for question 4: 12 marks

End of questions