A particle moves along the x-axis obeying the equation `x=t(t-1)(t-2)`, where x is in meter and t is in second
a. Find the initial velocity of the particle.
b. Find the initial acceleration of the particle.
c. Find the time when the displacement of the particle is zero.
d. Find the displacement when the velocity of the particle is zero.
e. Find the acceleration of the particle when its velocity is zero.

Explanation

`x=t[t-1][t-2]=(t^2-t)(t-2)=t^3-2t^2-t^2+2t=t^3-3t^2+2t`
`v=(dx)/(dt)=3t^2-6t+2`
a. `v_(t=0)=3xx0^2-6xx0+2=2ms^-1`
b. `a=(dv)/(dt)=6t-6`, `a_(t=0)=-6ms^-2`
c. `x=0impliest=0s`, `t=1s`, `t=2s`
d. `v=0`
`implies3t^2-6t+2=0`
`impliest=(6+-sqrt(6^2-4xx3xx2))/(6)=(6+-sqrt12)/(6)`
`1+-1/sqrt3=(sqrt3+-1)/(sqrt3)s`
At `t=(sqrt3-1)/(sqrt3)s`, `x=((sqrt-1)/(sqrt3))((sqrt-1)/(sqrt3)-1)((sqrt3-1)/(sqrt3)-2)`
`=(2)/(3sqrt3)m`
and at `t=(sqrt3+1)/(sqrt3)s`,
`x=((sqrt3+1)/(sqrt3))((sqrt+1)/(sqrt3)-1)((sqrt3+1)/(sqrt3)-2)=(-2)/(3sqrt3)m`
e. `a=6t-6=6((sqrt3-1)/(sqrt3))-6=-(6)/(sqrt3)=-2sqrt3ms^-2`
and `a=6((sqrt3+1)/(sqrt3))-6=6/sqrt3=2sqrt3ms^-1`