Example 7
A particle moves along a straight line so that its displacement, s meters, after passing through a fixed point O is given by s=t3 -9t2+24t+5, where t is the time in seconds, after the movement is started. Calculatea) the initial velocity, in ms-1 ,of the particle,b) the instantaneous velocity, in ms-1,at 3 seconds,c) the values of t, in seconds, when the particle is instantaneously at rest,d) the range of t, in seconds, when the particle moves to the left
Solution
Given displacement function s =t3- 9t2+24t+5,and velocity function, v==3t2-18t+24
a) When initial velocity,t=0,v=3(0)2-18(0)+24
v=24
Hence, the initial velocity of the particle is 24 ms-1.b) When t=3,v=3(3)2-18(3)+24=27-54+24=-3
Hence, the velocity at 3 seconds is -3ms-1.
c) When the particle is at rest, v = 0.
3t2-18t+24=0
t2-6t+8=0
(t – 2)(t – 4) = 0
t=2 or t=4
Hence, the particle rests instantaneously at 2s or 4s.
d) When particle moves to left, v<0
3t2-18t+24<0
t2-6t+8<0
(t-2)(t-4)<0
From the graph, the solution is 2< t < 4.
Hence, the particle moves to the left when 2< t < 4.