The Relationship among Position, Velocity and Acceleration

Position, velocity and acceleration are concepts all of us deal with every day. We give directions to a friend about our location in town in order to meet up at a restaurant. We drive to work as we push the acceleration and brake pedals in order to change the velocity of the car. These are just every day examples of how these concepts affect our life. We do not give much thought to these actions, but at the mathematical level there is more than just doing our daily activities.

The relationship among position, velocity and acceleration is one of the fundamental applications of calculus. From the point of view of calculus, these quantities are related by differentiation and integration. This tells us that if we know at least one of them, we can calculate the other two. The figure below shows the relationship among these quantities.

The arrows in the figure above show the directions in which we have to either differentiate or integrate in order to obtain the other quantities. A simple example is shown next.

Solution:

Differentiate the position equation to obtain the velocity. Then evaluate at t = 10.

Differentiate the velocity equation to obtain the acceleration. Then evaluate at t = 10.

At t = 10, the velocity and acceleration are 97 and 10, respectively, in their corresponding units.

Derivation of Free Fall Equations

Rectilinear motion with constant acceleration is introduced early in physics courses. While there are many cases of this type of motion, we can consider free fall one of the best examples of the relationship among position, velocity and acceleration (click here). It is even more important, since it shows how physics and calculus go hand in hand. Therefore, we will observe how to derive three of the equations for this special case.

Free fall is simply the motion of a falling object under the influence of acceleration due to gravity, g, which is considered a constant. Therefore, we will start with the acceleration function and then apply integration to get the velocity and position as functions of time. The procedure is shown below:

Finally, we rewrite the equations in the most common form that appears in most textbooks for consistency.