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
= t^{2} +
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
t^{3} + 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.
