1. (2011/june/Paper02/q2)
A particle is moving along a straight line. At time t seconds, t⩾, 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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