AP® Physics C: Mechanics · Unit 1 · Topic 1.2
Recovering Motion by Integration
If you know acceleration as a function of time plus the starting velocity and position, you can integrate twice to rebuild the entire motion. Each integration needs one initial condition to pin down its constant.
Velocity is the antiderivative of acceleration, and position is the antiderivative of velocity. That's just the reverse of differentiation: if a(t) = dv/dt, then v(t) = integral of a(t) dt. But integrating only gets you a family of curves that differ by a constant, written +C. Without more information, you don't know which member of that family actually describes your object. That's where initial conditions come in: the starting velocity v0 tells you which C makes v(t) correct, and the starting position x0 does the same job one step later for x(t).
Mechanically, you integrate a(t) with respect to t, get v(t) = (antiderivative) + C, then plug in t = 0 and set the result equal to v0 to solve for C. Repeat the process on v(t) to get x(t), using x0 to fix the new constant. Two integrations, two conditions, one complete description of the motion: where the object is and how fast it's going at any time you choose.
This is the natural follow-up to differentiating position to get velocity and acceleration from 1.1. There you tear a motion apart; here you rebuild it. The same idea scales up to two dimensions, where you integrate x and y components of acceleration separately, each with its own pair of initial conditions, to recover the full path of motion.
Key terms
- Antiderivative
- A function whose derivative gives back the original function; integration is the process of finding it.
- Constant of integration
- The unknown added term (+C) left over after integrating, which must be solved using a known condition.
- Initial condition
- A known value, like position or velocity at t = 0, used to fix the constant of integration.
- Definite vs indefinite integral
- An indefinite integral gives a function plus C; a definite integral gives a specific number over set time limits.
Watch it explained
- How it worksHow does integrating acceleration with respect to time hand back the velocity function?
- Worked exampleFind v(t) and x(t) for a particle with a(t) = 6t m/s2 that starts at rest from x = 2 m.
- Worked exampleFind the position at t = 4 s of an object with v(t) = 3t2 - 4t m/s that started at the origin.
- Common mistakeWhy isn't the constant of integration just bookkeeping you are allowed to drop?
Check yourself
End of topic.