Final answer:
To find the velocity function v(t), integrate and use initial conditions to find the constant of integration. Set the velocity function equal to zero and solve for t to find the time at which the velocity is zero. To find the initial position, integrate and evaluate the position function at the time when the velocity is zero.
Step-by-step explanation:
To find the velocity function v(t), we integrate and use initial conditions to find the constant of integration. At t = 0, we have v(0) = 5.0 m/s, so the constant of integration C₁ is 5.0 m/s. Therefore, the velocity function is v(t) = 5.0 m/s - t² m/s³.
Next, we set the velocity function equal to zero and solve for t to find the time at which the velocity is zero. We have 0 = 5.0 m/s - t² m/s³, which gives t = 6.3 s.
To find the initial position, we integrate and use the fact that x(0) = 0. Since the initial position is taken to be zero, we only have to evaluate the position function at the time when the velocity is zero. Therefore, at t = 6.3 s, the position function is x(t) = ¾/₁² – 1³.