Variable acceleration in one dimensionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A particle $P$ moves in a straight line. At time $t$ seconds, where $t\ge0$, its acceleration is $(6t-12)$ m s⁻² in the positive direction. At $t=0$, $P$ is at the origin $O$ with velocity 9 m s⁻¹ in the positive direction.

   (a)

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

   [1 mark]

   • A$-12$ m s⁻¹
   • B$0$ m s⁻¹
   • C$-3$ m s⁻¹
   • D$3$ m s⁻¹

   (b)

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

   [1 mark]

   • A$2$ m
   • B$-3$ m
   • C$-6$ m
   • D$-16$ m

   (c)

   Find the times at which $P$ is instantaneously at rest.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A sports car accelerates from rest along a straight, level track. At time $t$ seconds its velocity is $v$ m s⁻¹ and its acceleration is $\frac{60-v}{20}$ m s⁻².

   (a)

   Which expression gives $v$ in terms of $t$?

   [1 mark]

   • A$60e^{-0.05t}$
   • B$60\left(1-e^{0.05t}\right)$
   • C$60\left(1-e^{-20t}\right)$
   • D$60\left(1-e^{-0.05t}\right)$

   (b)

   Find the time taken for the car to reach 30 m s⁻¹.

   [1 mark]

   • A$20.0$ s
   • B$13.9$ s
   • C$27.7$ s
   • D$0.693$ s

   (c)

   Find the time taken for the car to reach 90% of its limiting speed.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves along the $x$-axis. At time $t$ seconds its velocity in the positive $x$-direction is $6\cos2t$ m s⁻¹. At $t=0$, $P$ is at the point with coordinate $x=1$. Angles are in radians.

   (a)

   Find an expression for $x$ in terms of $t$.

   [3 marks]

   (b)

   Find the acceleration of $P$ when $t=\frac{\pi}{3}$, and the greatest value of $x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A sledge slides in a straight line across ice and is modelled as a particle. At time $t$ seconds its speed is $v$ m s⁻¹ and its deceleration is $0.02v^2$ m s⁻². At $t=0$ its speed is 10 m s⁻¹.

   (a)

   Show that $v=\frac{10}{1+0.2t}$, and find the speed of the sledge when $t=10$.

   [6 marks]

   (b)

   Find the distance travelled by the sledge in the first 10 seconds, and explain why, in this model, the sledge never comes to rest.

   [6 marks]

   Total for question 4: 12 marks

End of questions