The acceleration of a particle is increasing linearly with time \[t\] as \[bt\]. The particle starts from the origin with an initial velocity \[{{v}_{0}}\] The distance travelled by the particle in time \[t\] will be [CBSE PMT 1995]

• A. \[{{v}_{0}}t+\frac{1}{3}b{{t}^{2}}\]     
• B. \[{{v}_{0}}t+\frac{1}{3}b{{t}^{3}}\]
• C. \[{{v}_{0}}t+\frac{1}{6}b{{t}^{3}}\]
• D.     \[{{v}_{0}}t+\frac{1}{2}b{{t}^{2}}\]