$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 21, question 9)
A particle moves in a straight line so that, $\D t$ s after passing through a fixed point $\D O,$ its velocity, $\D v$ ms$\D^{-1},$ is given by $\D v = 2t - 11 +\frac{6}{t+1}.$ Find the acceleration of the particle when it is at instantaneous rest.
[7]
2 (CIE 2012, w, paper 13, question 7)
A particle $\D P$ moves along the x-axis such that its distance, $\D x$ m, from the origin $\D O$ at time $\D t$ s is given by $\D x = \frac{t}{t^2+1}$ for $\D t\ge 0.$(i) Find the greatest distance of $\D P$ from $\D O.$
[4]
(ii) Find the acceleration of $\D P$ at the instant when $\D P$ is at its greatest distance from $\D O.$
[3]
3 (CIE 2012, w, paper 21, question 11either)
A particle travels in a straight line so that, $\D t$ s after passing through a fixed point $\D O,$ its displacement, $\D s$ m, from $\D O$ is given by $\D s = t^2 - 10t + 10\ln(l + t),$ where $\D t > 0.$(i) Find the distance travelled in the twelfth second.
[2]
(ii) Find the value of $\D t$ when the particle is at instantaneous rest.
[5]
(iii) Find the acceleration of the particle when $\D t = 9.$
[3]
4 (CIE 2012, w, paper 21, question 11or)
A particle travels in a straight line so that, $\D t$ s after passing through a fixed point $\D O,$ its velocity, $\D v$ cms$\D^{-1},$ is given by $\D v = 4e^{2t} - 24t.$(i) Find the velocity of the particle as it passes through $\D O.$
[1]
(ii) Find the distance travelled by the particle in the third second.[4]
(iii) Find an expression for the acceleration of the particle and hence find the stationary value of the velocity.
[5]
5 (CIE 2012, w, paper 22, question 10)
The acceleration, $\D a$ m s$\D^{-2},$ of a particle, $\D t$ s after passing through a fixed point $\D O,$ is given by $\D a = 4 - 2t,$ for $\D t > 0.$ The particle, which moves in a straight line, passes through $\D O$ with a velocity of 12 m s$\D^{-1}.$
(i) Find the value of $\D t$ when the particle comes to instantaneous rest. [5]
(ii) Find the distance from $\D O$ of the particle when it comes to instantaneous rest. [3]
6 (CIE 2013, s, paper 12, question 12)
A particle P moves in a straight line such that, $\D t$ s after leaving a point $\D O,$ its velocity $\D v$ m s$\Delta^{-1}$ is given by $\D v = 36t -3t^2$ for $\D t>0.$
(i) Find the value of $\D t$ when the velocity of $\D P$ stops increasing. [2]
(ii) Find the value of $\D t$ when $\D P$ comes to instantaneous rest. [2]
(iii) Find the distance of $\D P$ from $\D O$ when $\D P$ is at instantaneous rest. [3]
(iv) Find the speed of $\D P$ when $\D P$ is again at $\D O.$ [4]
7 (CIE 2013, w, paper 23, question 9)
A particle travels in a straight line so that, $\D t$ s after passing through a fixed point $\D O,$ its velocity, $\D v$ ms$\D^{-1},$ is given by $\D v = 3 + 6 \sin 2t .$
(i) Find the velocity of the particle when $\D t=\frac{\pi}{4}.$ [1]
(ii) Find the acceleration of the particle when $\D t = 2.$ [3]
The particle first comes to instantaneous rest at the point $\D P.$
(iii) Find the distance $\D OP.$ [5]
8 (CIE 2014, s, paper 13, question 8)
A particle moves in a straight line such that, t s after passing through a fixed point $\D O,$ its velocity, $\D v$ ms$\D^{-1} ,$ is given by $\D v = 5 - 4e^{-2t}.$
(i) Find the velocity of the particle at $\D O.$ [1]
(ii) Find the value of $\D t$ when the acceleration of the particle is 6ms$\D^{-2} .$ [3]
(iii) Find the distance of the particle from $\D O$ when $\D t = 1.5.$ [5]
(iv) Explain why the particle does not return to $\D O.$ [1]
9 (CIE 2014, w, paper 21, question 7)
A particle moving in a straight line passes through a fixed point $\D O.$ The displacement, $\D x$ metres, of the particle, $\D t$ seconds after it passes through $\D O,$ is given by $\D x = t + 2 \sin t.$
(i) Find an expression for the velocity, $\D v$ms$\D^{-1} ,$ at time $\D t.$ [2] When the particle is first at instantaneous rest, find
(ii) the value of $\D t,$ [2]
(iii) its displacement and acceleration. [3]
10 (CIE 2014, w, paper 23, question 8)
A particle moving in a straight line passes through a fixed point $\D O.$ The displacement, $\D x$ metres, of the particle, $\D t$ seconds after it passes through $\D O,$ is given by $\D x = 5t - 3 \cos 2t + 3.$
(i) Find expressions for the velocity and acceleration of the particle after $\D t$ seconds. [3]
(ii) Find the maximum velocity of the particle and the value of $\D t$ at which this first occurs. [3]
(iii) Find the value of $\D t$ when the velocity of the particle is first equal to 2 ms$\D^{-1}$ and its acceleration at this time. [3]
11 (CIE 2015, s, paper 12, question 6)
A particle moves in a straight line such that its displacement, $\D x$ m, from a fixed point $\D O$ is given by $\D x= 10 \ln(t^2+4)-4t.$
(i) Find the initial displacement of the particle from $\D O.$ [1]
(ii) Find the values of $\D t$ when the particle is instantaneously at rest. [4]
(iii) Find the value of $\D t$ when the acceleration of the particle is zero. [5]
12 (CIE 2015, s, paper 21, question 6)
A particle $\D P$ is projected from the origin $\D O$ so that it moves in a straight line. At time $\D t$ seconds after projection, the velocity of the particle, $\D v$ ms$\D^{-1},$ is given by $\D v= 2t^2-14t+12.$
(i) Find the time at which $\D P$ first comes to instantaneous rest. [2]
(ii) Find an expression for the displacement of $\D P$ from $\D O$ at time $\D t$ seconds. [3]
(iii) Find the acceleration of $\D P$ when $\D t = 3.$ [2]
13 (CIE $2015, \mathrm{w}$, paper 21 , question 10)
A particle is moving in a straight line such that its velocity, $v \mathrm{~ms}^{-1}, t$ seconds after passing a fixed point $O$ is $v=\mathrm{e}^{2 t}-6 \mathrm{e}^{-2 t}-1$
(i) Find an expression for the displacement, $s \mathrm{~m}$, from $O$ of the particle after $t$ seconds.$[3]$
(ii) Using the substitution $u=\mathrm{e}^{2 t}$, or otherwise, find the time when the particle is at rest.[3]
(iii) Find the acceleration at this time. [2]
14 (CIE 2015, w, paper 23, question 7)
The velocity, $v \mathrm{~ms}^{-1}$, of a particle travelling in a straight line, $t$ seconds after passing through a fixed point $O$, is given by $v=\frac{10}{(2+t)^{2}}$.
(i) Find the acceleration of the particle when $t=3$.
(ii) Explain why the particle never comes to rest.[1]
(iii) Find an expression for the displacement of the particle from $O$ after time $t \mathrm{~s}$. [3]
(iv) Find the distance travelled by the particle between $t=3$ and $t=8$.[2]
15 (CIE 2016, march, paper 22, question 12)
A particle $P$ is projected from the origin $O$ so that it moves in a straight line. At time $t$ seconds after projection, the velocity of the particle, $v \mathrm{~ms}^{-1}$, is given by $=9 t^{2}-63 t+90$
(i) Show that $P$ first comes to instantaneous rest when $t=2$.[2]
(ii) Find the acceleration of $P$ when $t=3.5$. [2]
(iii) Find an expression for the displacement of $P$ from $O$ at time $t$ seconds.[3]
(iv) Find the distance travelled by $P$
(a) in the first 2 seconds,[2]
(b) in the first 3 seconds. [2]
16 (CIE $2016, \mathrm{w}$, paper 11, question 10$)$
(a)The diagram shows part of the velocity-time graph for a particle, moving at $v \mathrm{~ms}^{-1}$ in a straight line, $t$ s after passing through a fixed point. The particle travels at $U \mathrm{~ms}^{-1}$ for $20 \mathrm{~s}$ and then decelerates uniformly for $10 \mathrm{~s}$ to a velocity of $\frac{U}{2} \mathrm{~ms}^{-1}$. In this 30 s interval, the particle travels $165 \mathrm{~m}$
(i) Find the value of $U$.[3]
(ii) Find the acceleration of the particle between $t=20$ and $t=30$.[2]
(b) A particle $P$ travels in a straight line such that, $t$ s after passing through a fixed point $O$, its velocity, $v \mathrm{~ms}^{-1}$, is given $v=\left(\mathrm{e}^{\frac{r^{2}}{8}}-4\right)^{3}$
(i) Find the speed of $P$ at $O$.[1]
(ii) Find the value of $t$ for which $P$ is instantaneously at rest.[2]
(iii) Find the acceleration of $P$ when $t=1$.[4]
17 (CIE 2016, w, paper 13, question 11)
A particle moving in a straight line has a velocity of $v \mathrm{~ms}^{-1}$ such that, $t$ s after leaving a fixed point, $v=4 t^{2}-8 t+3$
(i) Find the acceleration of the particle when $t=3$. [2]
(ii) Find the values of $t$ for which the particle is momentarily at rest.[2]
(iii) Find the total distance the particle has travelled when $t=1.5$.[5]
18 (CIE 2017, s, paper 12, question 5)
A particle $P$ moves in a straight line, such that its displacement, $x \mathrm{~m}$, from a fixed point $O, t$ s after passing $O$, is given by $x=4 \cos (3 t)-4$.
(i) Find the velocity of $P$ at time $t$.[1]
(ii) Hence write down the maximum speed of $P$. [1]
(iii) Find the smallest value of $t$ for which the acceleration of $P$ is zero. [3]
(iv) For the value of $t$ found in part (iii), find the distance of $P$ from $O$. [1]
19 (CIE 2017, s, paper 13, question 12) A particle moves in a straight line, such that its velocity, $v \mathrm{~ms}^{-1}, t$ s after passing a fixed point $O$, is given by $v=2+6 t+3 \sin 2 t$.
(i) Find the acceleration of the particle at time $t$. [2]
(ii) Hence find the smallest value of $t$ for which the acceleration of the particle is zero. [2]
(iii) Find the displacement, $x \mathrm{~m}$ from $O$, of the particle at time $t$.$[5]$
20 (CIE 2017, s, paper 21, question 12)
A particle moves in a straight line so that, $t$ seconds after passing a fixed point $O$, its displacement, $s \mathrm{~m}$, from $O$ is given by $s=1+3 t-\cos 5 t$
(i) Find the distance between the particle's first two positions of instantaneous rest. [7]
(ii) Find the acceleration when $t=\pi$.[2]
21 (CIE 2017, w, paper 23 , question 7) A particle moving in a straight line passes through a fixed point $O$. Its velocity, $v \mathrm{~ms}^{-1}, t \mathrm{~s}$ after passing through $O$, is given by $v=3 \cos 2 t-1$ for $t \geqslant 0$.
(i) Find the value of $t$ when the particle is first at rest.[2]
(ii) Find the displacement from $O$ of the particle when $t=\frac{\pi}{4}$.[3]
(iii) Find the acceleration of the particle when it is first at rest.[3]
22 (CIE 2018, march, paper 12, question 8)
A particle $P$, moving in a straight line, passes through a fixed point $O$ at time $t=0 \mathrm{~s}$. At time $t \mathrm{~s}$ after leaving $O$, the displacement of the particle is $x \mathrm{~m}$ and its velocity is $v \mathrm{~ms}^{-1}$, where $v=12 \mathrm{e}^{2 t}-48 t, t \geqslant 0$
(i) Find $x$ in terms of $t$.[4]
(ii) Find the value of $t$ when the acceleration of $P$ is zero.[3]
(iii) Find the velocity of $P$ when the acceleration is zero.[2]
23 (CIE 2018, s, paper 12, question 4)
A particle $P$ moves so that its displacement, $x$ metres from a fixed point $O$, at time $t$ seconds, is given by $x=\ln (5 t+3)$.
(i) Find the value of $t$ when the displacement of $P$ is $3 \mathrm{~m}$.[2]
(ii) Find the velocity of $P$ when $t=0$.[2]
(iii) Explain why, after passing through $O$, the velocity of $P$ is never negative.[1]
(iv) Find the acceleration of $P$ when $t=0$.[2]
24 (CIE 2018, s, paper 21, question 10)
A particle moves in a straight line such that its displacement, $s$ metres, from a fixed point $O$ at time $t$ seconds, is given by $s=4+\cos 3 t$, where $t \geqslant 0$. The particle is initially at rest.
(i) Find the exact value of $t$ when the particle is next at rest.[2]
(ii) Find the distance travelled by the particle between $t=\frac{\pi}{4}$ and $t=\frac{\pi}{2}$ seconds.[3]
(iii) Find the greatest acceleration of the particle.$[2]$
Answers
1. 11/6
2. (i)1/2(ii)a = -0.5
3. (i)13.8(ii)t = 4(iii)1.9
4. (i)v = 4(ii)638(iii)-1.18
5. (i)t = 6 (ii)s = 72
6. (i)t = 6 (ii)t = 12
(iii)s = 864 (iv)324
7. (a)(i)9 (ii)-7.84(iii)11.1
8. (i)1; t = 0.144; s = 5.6
(iv)V is always positive
9. (i)v = 2 cos t + 1
(ii)t = 2.09
(iii) $\D t =-\sqrt{3}$
10. (i)v = 5 + 6 sin 2t
a = 12 cos 2t
(ii) $\D t=\frac{\pi}{4}$; v = 11
(iii) $\D t=\frac{7\pi}{12}; a = -6\sqrt{3}$
11. (i)10 ln 4 (ii)t = 1; 4
(iii)t = 2
12. (i)1
(ii) $\D 2t^3/3-14t^2/2+12t$
(iii)-2
13. (i) $s=.5 e^{2 t}+3 e^{-2 t}-t-3.5$
(ii) $\ln \sqrt{3}$, (iii) 10
14. (i) $a=-0.16$
(ii) $10 /(t+2)^{2}$ is never zero
(iii) $s=-10 /(t+2)+5(\mathrm{iv}) 1$
15. (ii) 0
(iii) $9 t^{3} / 3-63 t^{2} / 2+90 t$
(iv)(a) 78, (b) $88.5$
16. (a)(i) $U=6$, (ii)-0.3
(bi) $27,($ ii $) 3.33($ iii $) 6.98$
17. (i) $16($ ii $) 0.5,1.5$
(ii) $4 / 3$
18. (i) $v=-12 \sin 3 t$
(ii) 12
(iii) $\pi / 6$
(iv) 4
19. (i) $6 \cos 2 t+6$
(ii) $t=\pi / 2$
(iii) $x=1.5-1.5 \cos 2 t+3 t^{2}+2 t$
20. (i) $0.48$, (ii) $-2.5$
21. (i) $0.615($ ii $) 0.715$ (iii) $-5.66$
22. (i) $x=6 e^{2 t}-24 t^{2}-6$
(ii) $(1 / 2) \ln 2$
(iii) $24-24 \ln 2$
23. (i) $t=3.42$
(ii) $1.67$
(iii) never negative
(iv) $-2.78$
24. (i) $t=\pi / 3$
(ii) $1.29$
(iii) 9
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