Further Mechanics 2: Further kinematicsEdexcel A-Level Further Maths: Topic test

1. 1
   A cyclist starts from rest at time $t=0$ and moves along a straight track. For $0\le t\le5$, where $t$ is in seconds, the acceleration of the cyclist is $(2-0.4t)$ m s⁻².

   (a)

   What is the speed of the cyclist when $t=5$?

   [1 mark]

   • A$10$ m s⁻¹
   • B$5$ m s⁻¹
   • C$2$ m s⁻¹
   • D$0$ m s⁻¹

   (b)

   What is the distance travelled by the cyclist in the first 5 seconds?

   [1 mark]

   • A$25$ m
   • B$12.5$ m
   • C$8.33$ m
   • D$16.7$ m

   (c)

   Find the time at which the speed of the cyclist is 4.2 m s⁻¹.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cart moves along a straight horizontal track. At time $t$ seconds its speed is $v$ m s⁻¹, and magnetic braking gives it a deceleration of $0.4v$ m s⁻². The initial speed of the cart is 10 m s⁻¹.

   (a)

   Which expression gives $v$ at time $t$?

   [1 mark]

   • A$10\mathrm e^{-0.4t}$
   • B$10-0.4t$
   • C$10\mathrm e^{0.4t}$
   • D$4\mathrm e^{-10t}$

   (b)

   After how many seconds is the speed of the cart 5 m s⁻¹?

   [1 mark]

   • A$12.5$ s
   • B$0.277$ s
   • C$1.73$ s
   • D$2.5$ s

   (c)

   Find the distance travelled by the cart in the first 2 seconds.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ moves along the $x$-axis. At time $t=0$, $P$ is at the origin $O$ and has velocity 4 m s⁻¹ in the positive $x$-direction. At time $t$ seconds, the acceleration of $P$ is $6\mathrm e^{-2t}$ m s⁻².

   (a)

   Show that the velocity of $P$ at time $t$ is $\left(7-3\mathrm e^{-2t}\right)$ m s⁻¹.

   [3 marks]

   (b)

   Find the distance of $P$ from $O$ when $t=\ln2$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A drone is launched from rest at ground level and rises vertically. At time $t$ seconds, where $t\ge0$, its upward acceleration is $6\left(1-t^2\right)\mathrm e^{-t^2/2}$ m s⁻².

   (a)

   Show that $\frac{\mathrm d}{\mathrm dt}\left(t\mathrm e^{-t^2/2}\right)=\left(1-t^2\right)\mathrm e^{-t^2/2}$. Hence find the velocity of the drone at time $t$, and its greatest speed.

   [6 marks]

   (b)

   Find the height of the drone above the ground at time $t$. Hence find the time at which the drone is 3 m above the ground.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A particle $P$ moves in a straight line. At time $t=0$ its speed is 4 m s⁻¹, and while it is moving in the positive direction its deceleration has magnitude $v^2$ m s⁻², where $v$ m s⁻¹ is its speed at time $t$ seconds.

   (a)

   Which expression gives $v$ at time $t$?

   [1 mark]

   • A$4\mathrm e^{-t}$
   • B$\frac{4}{1+t}$
   • C$4-16t$
   • D$\frac{4}{1+4t}$

   (b)

   At what time is the speed of $P$ equal to 0.5 m s⁻¹?

   [1 mark]

   • A$7$ s
   • B$1.75$ s
   • C$0.4375$ s
   • D$2$ s

   (c)

   Find the distance travelled by $P$ in the first second.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A lift moves vertically for 4 seconds. At time $t$ seconds, where $0\le t\le4$, its upward velocity is $2\sin\left(\frac{\pi t}{4}\right)$ m s⁻¹.

   (a)

   What is the acceleration of the lift when $t=0$?

   [1 mark]

   • A$\frac{\pi}{4}$ m s⁻²
   • B$2$ m s⁻²
   • C$\frac{\pi}{2}$ m s⁻²
   • D$\frac{8}{\pi}$ m s⁻²

   (b)

   What is the distance travelled by the lift in the 4 seconds?

   [1 mark]

   • A$\frac{16}{\pi}$ m
   • B$\frac{8}{\pi}$ m
   • C$8$ m
   • D$4$ m

   (c)

   Find the value of $t$, with $0<t\le4$, at which the acceleration of the lift is zero.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ starts from rest at time $t=0$ and moves in a straight line. At time $t$ seconds, where $t\ge0$, the acceleration of $P$ is $\frac{12}{(1+t)^2}$ m s⁻² in the direction of motion.

   (a)

   Show that the speed of $P$ at time $t$ is $\frac{12t}{1+t}$ m s⁻¹.

   [3 marks]

   (b)

   Find the distance travelled by $P$ in the first 5 seconds.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle $P$ moves along the positive $x$-axis. At time $t=0$, $P$ is at the origin $O$ with speed 2 m s⁻¹. At time $t$ seconds, the acceleration of $P$ is $\frac3v$ m s⁻², where $v$ m s⁻¹ is the speed of $P$ and $v>0$.

   (a)

   Show that $v=\sqrt{6t+4}$.

   [6 marks]

   (b)

   Find the distance of $P$ from $O$ when $t=10$, and the acceleration of $P$ at that time.

   [6 marks]

   Total for question 8: 12 marks

End of questions