This equation can be derived by combining two equations for average velocity. Average velocity is defined as displacement divided by the time during which that displacement occurred:

So this really boils down to thinking about when is the velocity positive or negative. And to think about that, let's actually graph the velocity function or make a rough sketch of it.

The definite integral of a velocity function gives us the displacement. To find the actual distance traveled, we need to use the speed function, which is the absolute value of the velocity.

Definite integrals are commonly used to solve motion problems, for example, by reasoning about a moving object's position given information about its velocity. Learn how this is done and …

Join us as we explore the relationship between position, velocity, and acceleration for a particle moving along the x-axis. We'll calculate velocity and acceleration from a given position …

Our velocity function, which is the derivative of the position function, is telling that story. Out the gate, we have a high positive velocity, but we decelerate quickly.

Well, you could just integrate the velocity function, which will give you a position function. Using that, you can find the position at t=1 (which would also be its displacement).

At what value of T does the particle obtain its maximum acceleration? So we want to figure out when does it obtain its maximum acceleration. So let's just review what they gave us. They …

In my Calculus book, there is a word problem which gives me the acceleration function, and one x and y point on the position function, but does not mention a point on the velocity function. …

And so the absolute value of the velocity function, this would give you, integrating the speed, this would give you the distance. Distance from t equals two to t is equal to six, and let's see, we …