A particle starts from rest at t=0 and undergoes an acceleration as shown in figure.
Draw the velocity-time graph of the motion of the particle during the interval from zero to 4 second.

Explanation

The displacement of the particle in time interval t = 0 to t = 2 is given by
`s=ut+(1)/(2)" at"^(2) =0+(1)/(2) xx3xxt^(2) (3)/(2) t^(2)`
Therefore, the displacement at t = 0, t = 1 s and t = 2 s respectively are 0, 1.5 m and 6 m. In the time interval from t = 2s to t = 4 or t . = 0 or t . = 2s ( t . = t – 2), the displacement is given by
`s.=ut.+(1)/(2) a(t.)^(2) =6t. (1)/(2) xx3xx t^(2) =6t-1.5 t^(2)`
(Hence for this time interval `u = 6 m//s and a = – 3 m//s^(2))`. The displacement at t . = 1s and t . = 2 i.e., t = 3s and t = 4s are, therefore, (6 –1.5) = 4.5 m and (12 – 6) = 6 m respectively more than the displacement at t = 2s which is 6 m . Hence the displacement at t = 3s and t = 4s are (4.5 + 6) = 10.5 cm and (6 + 6) = 12 m respectively.
These results are tabulated below:
Time (s) 0 1 2 3 4
Displacement (m/s) 0 1.5 6 10.5 12
Figure shows the displacement-time graph of the motion