Wednesday, December 12, 2018

Kinematics (Application in Differentation)

$\def\D{\displaystyle}$

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]

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. (ai)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

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