. Find when and where the point first comes to rest, and its acceleration then.
V. VELOCITY AND ACCELERATION IN LINEAR MOTIONS
1. (i) A particle is projected vertically upwards, the distance x from the point O at time t is given by x = 5t(8-t). Find the initial velocity and the greatest height reached by the particle.
(ii) For a point moving along the x-axis, the displacement x at time t is given by . Find when and where the point first comes to rest, and its acceleration then.
2. For a particle moving along the line x'Ox the velocity v = dx/dt at the end of time t is given by
(i) , and when t=1, x=1/3; find x when t=2.
(ii) 
and when t=p/4, x=2; find x when t=3p/4.
(iii) 
and when t=0, x=2; find x when t=1.
3. The velocity of a particle moving along a straightline after time t is given by:
(i) 
and when t=1, x=3; find x when t=0.
(ii) 
and when t=1, x=3; find x when t=0.
4. (i) Write down the formula for cos(x+y) and hence show that 
. The velocity v of a particle moving along the x-axis is given by 
. If x=p/2 when t=p, find x when t=p/4.
(ii) The velocity v of a body moving in a straight line at time t is given by 
. Show that the body is momentarily at rest at each of two points A and B. And find the distance AB.
5. The velocity dx/dt of a particle at time t is given by 
. If the particle is travelling on the x-axis, find the
(i) average velocity over the interval 0<t<2;
(ii) acceleration when t=1;
(iii) instantaneous velocity when t=2.
6. If Va, Vb, Vc are the velocities of three particles A, B and C (moving along the x-axis) at time t , then 
. Assuming that in each case, when t=0, x=2, find their distance from the origin when t=1; which particle has travelled furthest.
7. For a particle moving along the x-axis, the acceleration 
at time t is given by:
(i) 
, and when t=0, 
, x=-3; find x when t=1.
(ii) 
and when t=1, v=2, x=11; find a when v=0, and the displacement of the particle over the time interval t=1 and t=e.
8. Prove that 
. A particle moves so that its acceleration at time t is given by 
(0<t<p/2). Given that when t=0, x=2, v=4 find an expression for x in terms of t.
9. For a particle moving along the x-axis, the velocity v at time t is given by 
. If when t=0, x=14, find
(i) the position of the particle at time t=9.
(ii) the time for it to move a distance of 19 units from the starting point and the corresponding time for it to move the same distance at its initial velocity.
(iii) the relation between the acceleration a and the velocity v.
10. For a particle moving along the x-axis, the acceleration at time t is given by 
. If, when t=0, x=0 , dx/dt=0, find x in terms of t. Give a sketch of x against t, for 0<t<p.
11. The equation of motion of a particle moving in a straightline is given by 
where k is a constant. The particle starts from rest at a point P distant 5 cm from a point O in the line and moves until the 6 sec, it reaches the point Q at which its velocity is 0. Calculate the (i) distance PQ, (ii) acceleration at Q, (iii) maximum velocity of the particle.
12. A particle starting from rest at O moves along a straightline x'Ox so that its acceleration at time t is 12t(2-t). Find
(i) when it again returns to O and its velocity then;
(ii) its maximum displacement from 0 during this interval;
(iii) its maximum positive velocity and its greatest speed during this interval (hints: draw a sketch of v against t).
13. A body moves in a straightline with initial velocity 9 cm/s. Its acceleration t sec after motion begins 2(4-t) cm/s2. Find how far the body moves before beginning to retrace its path, and prove that the elapsed time from the beginning of the motion before the particle returns to its starting point is 
sec.