When integrating the acceleration to determine the velocity and position, the initial conditions must be specified. The lower limits of integration will be

     v_o = 50 m/s      t_o = 2 s

When the acceleration is integrated, the expression for the velocity, as a function of time, becomes

     
            = t² + 46

     v(t = 4) = 62 m/s

The next step is to integrate the velocity. However, it is important that v(t) is integrated and not the velocity at 2 seconds, 62 m/s. This is a common mistake. Integrating the velocity function, v(t), gives an expression for the position as a function of time,

     
            = 0.333 t³ + 46 t - 94.67

     x(t = 4) = 110.7 m

Note that the initial position x_o was set to zero when t_o = 2 so that the equation for x(t) gives the change in position starting at t = 2.