# HOTS : Kinematics

Problem 1:  A particle moves along X-axis obeying the equation x = t (t – 1) (t – 2) where x is in metre &  t in second.  Find the (a) initial velocity and acceleration of the particle (b) velocity & acceleration of the particle when its displacement is zero (c) displacement & acceleration of the particle when its velocity is zero.

Solution :

(a)  x = t (t –1) (t – 2)                     . . . (i)

⇒  x = t3 – 3t2 + 2t

⇒ v = dx/dt  = 3t2 – 6t + 2               . . . (ii)

⇒ At time t = 0 , v = 2 m/sec.  &

a = dv/dt = 6t – 6 = 6(t –1)        . . . (iii)

⇒ at time t = 0,  a = -6m/sec2  .

(b) Displacement of the particle is zero at time t given by

x = t (t – 1) (t−2) = 0

⇒    t = 0 , t = 1 & t = 2

putting the values of t in eq. (ii) & (iii) we obtain,

velocity v = +2 , -1 & +2 m/sec.  respectively.

Acceleration a = −6 , 0 & +6 m/sec2 respectively.

(c) velocity v = 0

⇒ v = 3t2 – 6t + 2 = 0

⇒ t = 1 + (1/√3)    & t = 1 − (1/√3)

By putting these values of t in eq. (i) we obtain,

x = (−2/3√3) & (2/3√3) m respectively & the corresponding  acceleration can be obtained

by putting the values of t in  equation (iii) given by

a = 2√3 m/sec2  & – 2√3 m/s2 respectively

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