AP® Calculus AB · Unit 8 · Topic 8.2
Recovering Position from Velocity
Position is the antiderivative of velocity, but the antiderivative alone is only correct up to a constant. An initial position pins down that constant so you get the exact position function.
Velocity tells you how position is changing at every instant, but not where you started. Two particles can have the exact same velocity function and still be in completely different places at every time t, because one started at s = 0 and the other started at s = 100. The velocity only locks in the shape of the motion and the amount of change between times, not the actual location.
To recover position, integrate velocity: s(t) = integral of v(t) dt. This gives you a whole family of antiderivatives, all differing by a constant C, since d/dt of any constant is 0. The initial position s(0) tells you which member of that family is the real one. Plug t = 0 into your general antiderivative, set it equal to the given s(0), and solve for C. Once C is fixed, s(t) is a specific, complete function you can evaluate at any time.
This is the same logic that shows up constantly in net-change problems later in the unit, like a tank filling with water. There, a rate function gets integrated to find total change, and a starting amount pins down the actual quantity in the tank, exactly the way s(0) pins down position here. Whenever you integrate a rate, you get change; you always need one known value to convert that change into an absolute amount.
Key terms
- Position function
- s(t), the location of a particle at time t.
- Velocity function
- v(t), the rate of change of position with respect to time.
- Antiderivative
- A function whose derivative gives back the original function, unique only up to an added constant.
- Initial condition
- A known value, like s(0), used to solve for the constant of integration.
Watch it explained
- How it worksHow does an initial position turn a velocity integral into a position function?
- Worked exampleFind the position at t = 4 for v(t) = 3t2 - 2t when the particle starts at s = 5.
- Common mistakeWhy isn't an antiderivative of velocity the position without more information?
- Ties togetherHow does adding an initial position here compare with adding a starting amount in a tank's net-change problem later on?
Check yourself
End of topic.