The Big Idea
In particle-motion problems, the three quantities you work with most often are position, velocity, and acceleration. They are connected by derivatives:
Integration simply reverses those relationships. If you know acceleration, you can integrate once to find velocity. Then integrate velocity to find position.
Think of integration as moving backward through the chain . Acceleration integrates to velocity; velocity integrates to position.
From Acceleration to Velocity
Because acceleration is the derivative of velocity, if you are given , integrate it to obtain :
There is one important complication: an indefinite integral produces a constant of integration. That constant represents the fact that many different velocity functions can have the same acceleration.
For example, if , then both and have the same derivative. An initial condition such as tells you which velocity function is correct.
From Velocity to Position
Once you know velocity, integrate again. Since velocity is the derivative of position:
Again, an initial position is usually needed to determine the constant of integration. If you know , substitute that value into your position equation to find the constant.
A More Efficient Method: Definite Integrals
On AP Calculus problems, you do not always need to find an explicit formula for velocity or position. The Fundamental Theorem of Calculus lets you use definite integrals directly.
This is especially useful when the problem gives acceleration and asks for velocity at a particular time. If you know the initial velocity, integrate acceleration from the initial time to the desired time.
Similarly, if velocity is known and you want displacement:
And if the initial position is known, you can find the actual position:
Worked Examples
A particle moves along the -axis with acceleration . At time , its velocity is and its position is . Find the position function.
A particle has acceleration . Its velocity at time is . Find its velocity at time .
A particle has velocity and position . Find its position at time .
A particle has acceleration . At time , the particle has position and velocity . Find the position at time .
Position, Displacement, and Distance Are Not the Same
Integration can tell you where a particle is, but AP Calculus questions often distinguish carefully between position, displacement, and total distance traveled.
Displacement is the signed change in position. It can be negative because motion in the negative direction counts negatively.
Total distance traveled cannot be found by simply integrating velocity if velocity changes sign. Instead, you must integrate the speed, which is the absolute value of velocity.
This distinction is one of the most common sources of errors in particle-motion questions.
When Acceleration Is Given as a Function of Time
If acceleration is given directly as a function of time, the process is systematic:
- Integrate to get .
- Use the initial velocity to determine the constant in .
- Integrate to get .
- Use the initial position to determine the second constant in .
- Evaluate the resulting position function at the requested time.
If the question only asks for a value such as , a definite integral is often faster than finding the entire velocity function. If it asks for a general expression for or , you will usually need to integrate and apply the initial conditions.
Common Mistakes
Acceleration is the derivative of velocity, not velocity itself. If you are given , you must integrate before you can find .
An indefinite integral requires a constant. When integrating acceleration to find velocity, use the initial velocity to determine that constant. When integrating velocity to find position, use the initial position.
If velocity changes sign, positive and negative areas cancel when you calculate displacement. For total distance traveled, split the interval at the times when velocity is zero and integrate on each interval.
An initial velocity determines the constant in the velocity function. An initial position determines the constant in the position function. Keep those two jobs separate.
Practice Problems
A particle has acceleration . If , find .
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A particle has velocity and . Find the position at .
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A particle has acceleration , with and . Find .
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A particle's velocity is negative throughout the interval . Explain the relationship between its displacement and total distance traveled.
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