Glossary
Acceleration
The rate of change of velocity with respect to time, indicating how an object's speed and/or direction are changing, denoted as a(t) = v'(t) = s''(t).
Example:
When a roller coaster rapidly speeds up from rest, it experiences significant positive acceleration.
Constant of Integration
An arbitrary constant (C) added to the result of an indefinite integral, representing the initial condition or value of the original function.
Example:
When integrating acceleration to find velocity, the constant of integration represents the initial velocity of the object.
Displacement
The net change in position of an object from its initial to its final position, calculated as the definite integral of the velocity function over a time interval.
Example:
If you walk 5 meters forward and then 3 meters backward, your displacement is 2 meters forward, regardless of the total path taken.
Distance Traveled
The total length of the path an object takes, calculated as the definite integral of the absolute value of the velocity function (speed) over a time interval.
Example:
If a squirrel runs 10 feet up a tree and then 10 feet down, its distance traveled is 20 feet, even though its displacement is zero.
Fundamental Theorem of Calculus
A theorem linking the concepts of differentiating a function and integrating a function, crucial for evaluating definite integrals and finding net change.
Example:
Using the Fundamental Theorem of Calculus, you can find the exact change in position of a particle by evaluating the definite integral of its velocity function over a time interval.
Position
The location of an object in space at a given time, typically denoted as s(t).
Example:
If your car's GPS shows you are at mile marker 100, that's your position s(t) at that moment.
Speed
The magnitude of velocity, representing how fast an object is moving without regard to direction. For one-dimensional motion, it's the absolute value of velocity, |v(t)|.
Example:
Even if a car is backing up (negative velocity), its speed is still a positive value, like 5 mph.
Velocity
The rate of change of position with respect to time, indicating both speed and direction, denoted as v(t) = s'(t).
Example:
A car traveling at 60 mph north has a specific velocity, not just speed.