How do we find out how the position of a object varies with time, when it is moving under the influence of constant acceleration?
Well we can start by using how the velocity depends on time. How does it? We just figured it out, it's eqn. 3.11. So how can we get from v to x? We use the definition of instantaneous velocity, eq. 3.3. As in the above, we just turn around this equation, in symbols:
So now we have a slightly more complicated differential equation to solve. Following the same line of reasoning as above, we ask: what function has as its derivative ? In other words, what is the anti-derivative of ? You can answer this question for both the at and the terms separately, and then just add the answers together. You easily see the correct anti-derivative is
Again we have some arbitrary constant D that we can't determine with the information given. We need yet another initial condition to determine what that constant is.
So we follow the same steps as above. We say that the position of the object at t=0 is given, call it . Then at t=0 eq. 3.13 becomes . So now we know D. It's just . Plugging this back into eq. 3.13 gives
So if you know the initial position, the initial velocity, and the acceleration, then you can determine the position of the object as a function of time.