Further Pure Math (Kinematic)

 1. (2011/june/Paper02/q2)

A particle is moving along a straight line. At time $t$ seconds, $t \geqslant 0$, the displacement, $s$ metres, of the particle from a fixed point of the line is given by $s=t^{3}+2 t^{2}-3 t+6$

Find the value of $t$ for which the particle is moving with velocity $12 \mathrm{~m} / \mathrm{s}$. (4)


2. (2012/june/paper02/q9)

The particle $M$ is moving along the straight line $P Q$ with a constant acceleration of $2 \mathrm{~m} / \mathrm{s}^{2}$. At time $t=0, M$ is at the point $P$ moving with velocity $6 \mathrm{~m} / \mathrm{s}$ towards $Q$.

(a) Find an expression for the velocity of $M$ at time $t$ seconds.

(b) Show that the displacement of $M$ from $P$ at time $t$ seconds is $\left(t^{2}+6 t\right)$ metres.

A second particle $N$ is moving along $P Q .$ The acceleration of $N$ at time $t$ seconds is $6 t \mathrm{~m} / \mathrm{s}^{2}$. At time $t=0, N$ is stationary at the point $P$.

(c) Find an expression for the velocity of $N$ at time $t$ seconds.

(d) Find an expression for the displacement of $N$ from $P$ at time $t$ seconds.

(e) Find the distance between $M$ and $N$ at time $t=5$ seconds.

(f) Find the value of $t, t>0$, when the two particles meet.


3. (2013/jan/Paper01/q5)

A particle $P$ moves along the $x$-axis. At time $t$ seconds $(t \geqslant 0)$ the velocity, $v \mathrm{~m} / \mathrm{s}$, of $P$ is given by $v=5 \cos 2 t$. Find

(a) the least value of $t$ for which $P$ is instantaneously at rest,

(b) the magnitude of the maximum acceleration of $P$. $(3)$

When $t=0, P$ is at the point $(2,0)$.

(c) Find the distance of $P$ from the origin when $P$ first comes to instantaneous rest.


4. (2013/june/paper02/q10)

$$\begin{gathered}\tan \theta=\frac{\sin \theta}{\cos \theta} \\\cos (A+B)=\cos A \cos B-\sin A \sin B\end{gathered}$$

A particle $P$ is moving along a straight line. At time $t$ seconds $(t \geqslant 0)$ the displacement, $s$ metres, of $P$ from a fixed point $O$ on the line is given by $s=\sqrt{3} \sin \frac{1}{2} t+\cos \frac{1}{2} t$

(a) Find the exact value of $s$ when $t=\frac{\pi}{3}$

(b) Find the exact value of $t$ when $P$ first passes through $O$.

The velocity of $P$ at time $t$ seconds is $v \mathrm{~m} / \mathrm{s}$.

(c) Find an expression for $v$ in terms of $t$.

(d) Show that $v=\cos \left(\frac{\pi}{6}+\frac{1}{2} t\right)$

(e) Find the exact value of $t$ for which $v=\frac{1}{2}$ when

(i) $0 \leqslant t<2 \pi$

(ii) $2 \pi \leqslant t<4 \pi$


5. (2014/jan/paper01/q9)

A particle $P$ moves in a straight line such that, at time $t$ seconds, its displacement, $s$ metres, from a fixed point $O$ of the line is given by $s=t^{3}-6 t^{2}+5 t$

Find

(a) the values of $t$ for which $P$ passes through $O$

(b) the speed of $P$ each time it passes through $O$

(c) the greatest speed of $P$ in the interval $0 \leqslant t \leqslant 5$ (4)


6. $(2014 /$ june $/$ paper01/q7)

[In this question all distances are measured in metres.]

A particle $P$ is moving along the $x$-axis. At time $t$ seconds, $P$ is at the point with coordinates $\left(x_{p}, 0\right)$, where $x_{p}=8-10 t+\frac{1}{3} t^{3}$

Find, in terms of $t$,

(a) the velocity of $P$ at time $t$ seconds,

(b) the acceleration of $P$ at time $t$ seconds.

A second particle $Q$ is also moving along the $x$-axis. At time $t$ seconds, the velocity of $Q$ is $v_{Q} \mathrm{~m} / \mathrm{s}$, where $v_{Q}=t^{2}-3 t+4$

At time $t=0, Q$ is at the origin and at time $t$ seconds $Q$ is at the point with coordinates $\left(x_{Q^{+}} 0\right)$

(c) Find $x_{Q}$ in terms of $t$.

The particles $P$ and $Q$ collide at time $T$ seconds, where $T<5$

(d) Find the value of $T$.


7. (2015/june/paper02/q5)

A particle $P$ moves in a straight line such that at time $t$ seconds, the displacement, $s$ metres, of $P$ from a fixed point $O$ on the line is given by

$$ s=t^{3}-5 t^{2}+6 t \quad t \geqslant 0 $$

(a) Find the values of $t(t>0)$ when $P$ passes through $O$. (3)

(b) Find the speed of $P$ when $t=1$

(c) Find the magnitude of the acceleration of $P$ at each of the times when it passes through $O$. (3)


8. (2016/june/paper02/q7)

A particle $P$ moves in a straight line so that, at time $t$ seconds $(t \geqslant 0)$, its velocity, $v \mathrm{~m} / \mathrm{s}$, is given by $v=3 t^{2}-4 t+7$ Find

(a) the acceleration of $P$ at time $t=2$

(b) the minimum speed of $P$. (3)

When $t=0, P$ is at the point $A$ and has velocity $V \mathrm{~m} / \mathrm{s}$.

(c) Write down the value of $V$. (1)

When $P$ reaches the point $B$, the velocity of $P$ is also $V \mathrm{~m} / \mathrm{s}$.

(d) Find the distance $A B$.


9. (2017/jan/paper01/q10)

A particle $P$ moves along the positive $x$-axis. At time $t$ seconds $(t \geqslant 0)$ the velocity, $v \mathrm{~m} / \mathrm{s}$, of $P$ is given by $v=t^{3}-4 t^{2}+5 t+1$

The acceleration of $P$ at time $t$ seconds is $a \mathrm{~m} / \mathrm{s}^{2}$

(a) Find an expression for $a$ in terms of $t$.

(b) Find the values of $t$ for which the magnitude of the acceleration of $P$ is instantaneously zero.

When $t=0$, the displacement of $P$ from the origin is $3 \mathrm{~m}$.

(c) Find the displacement of $P$ from the origin when $t=2$


10. (2017/june/paper02/q4)

A particle $P$ is moving along a straight line which passes through the point $O$. At time $t=0$ the particle $P$ is at the point $O$.

At time $t$ seconds the velocity, $v \mathrm{~m} / \mathrm{s}$, of $P$ is given by $v=3 t^{2}+2 t+5$

(a) Find the acceleration of $P$ when $t=2$

(b) Find the displacement of $P$ from $O$ when $t=3$


11. (2018/jan/paper01/q4)

A particle $P$ moves along the $x$-axis. At time $t$ seconds $(t \geqslant 0)$, the displacement of $P$ from the origin is $x$ metres and the velocity, $v \mathrm{~m} / \mathrm{s}$, of $P$ is given by $v=2 t^{2}-16 t+30$

(a) Find the times at which $P$ is instantaneously at rest.

(b) Find the acceleration of $P$ at each of these times,

When $t=0, P$ is at the point where $x=-4$

(c) Find the distance of $P$ from the origin when $P$ first comes to instantaneous rest


12. (2018/june/paper01/q7)

A particle $P$ moves along the $x$-axis so that at time $t$ seconds, $t \geqslant 0$, the velocity of $P$, $v \mathrm{~m} / \mathrm{s}$, is given by $v=5 \cos 2 t$

(a) Find the value of $t$ when $P$ first comes to instantaneous rest.

(b) Find the magnitude of the maximum acceleration of $P$.

When $t=0, P$ is at the point $A$, where $O A=0.2 \mathrm{~m}$.

When $P$ first comes to instantaneous rest, $P$ is at the point $B$.

(c) Find the distance $O B$. (4)


13. ( $2019 /$ june $/$ paper02/q3)

A particle $P$ moves in a straight line. At time $t$ seconds, the velocity, $v \mathrm{~m} / \mathrm{s}$, of $P$ is given by

$$v=t^{2}-4 t+7$$

(a) Find the acceleration of $P$, in $\mathrm{m} / \mathrm{s}^{2}$, when $t=3$

(b) Find the distance, in $\mathrm{m}$, that $P$ travels in the interval $0 \leqslant t \leqslant 6$ (4)


14. (2019/juneR/paper02/q4)

A particle $P$ moves along the $x$-axis. At time $t$ seconds $(t \geqslant 0)$ the acceleration, $a \mathrm{~m} / \mathrm{s}^{2}$, of $P$ is given by $a=6 t-12$

When $t=0, P$ is at rest at the origin.

(a) Find the velocity of $P$ when $t=2$

At time $T$ seconds, $T>0, P$ is instantaneously at rest.

(b) Find the value of $T$. (2)

(c) Find the distance travelled by $P$ in the first 8 seconds of its motion.



Answer

1. $t=\frac{5}{3}$

2.(a) $v=2 t+6$ (b) Show (c) $v=3 t^{2}$ (d) $s=t^{3}$ (e) 70 (f) $t=3$

3.(a) $\quad t=\frac{\pi}{4}$ (b) $a_{\max }=10 \mathrm{~m} / \mathrm{s}^{2}$ (c) $4.5 \mathrm{~m}$

4.(a) $\sqrt{3}$ (b) $t=\frac{5 \pi}{3}$ (c) $v=\frac{\sqrt{3}}{2} \cos \frac{t}{2}-\frac{1}{2} \sin \frac{t}{2}$ (d) Show (e) (i) $t=\frac{\pi}{3}$ (ii) $t=3 \pi$

5.(a) $\quad t=0,1,5$ (b) $5,4,20$ (c) $20 \mathrm{~m} / \mathrm{s}$

6.(a) $v_{p}=-10+t^{2}(b) a_{p}=2 t$ (c) $x_{Q}=\frac{1}{3} t^{3}-\frac{3}{2} t^{2}+4 t$ (d) $T=0.61$

7.(a) $t=2,3$ (b) $1 \mathrm{~m} / \mathrm{s}$ (c) $t=2, a=2 ; t=3, a=8$

8.(a) $a=8$ (b) $5 \frac{2}{3}$ (c) $V=7$ (d) $A B=8 \frac{4}{27}$

9. (a) $a=3 t^{2}-8 t+5$(b) $t=1, \frac{5}{3}$(c) $s=8 \frac{1}{3}$

10. (a) $a=14$ (b) $s=51$

11.(a) $t=3,5$ (b) $t=3,a=-4;t=5,a=4$ (c) $s=32$

12.(a) $t=\frac{\pi}{4}$ (b) $a_{\max}=10$ (c) 2.7

13.(a) $2 \mathrm{~m} / \mathrm{s}^{2}$ (b) $42 \mathrm{~m}$

14.(a) $v=-12 m/s$ (b) $T=4$ (c) $d=192$(m)


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