Basic Kinematic Relationships

The study of kinematics is the study of the relationships between different elements of motion like position, velocity, and acceleration. Kinematics does not consider the forces that cause these motions, only the motions themselves. In this discussion I will use the symbol s for position, v for velocity, and a for acceleration. If a body travels from one position to another position and that trip starts at one time and ends at another time, we can write the average velocity for the body as:

However realize that this is the average velocity. The velocity at any point during the trip can vary because we are only considering the beginning and ending positions and times. To find the instantaneous velocity we use the same theory but we use calculus to express the change in position with respect to time using differential elements.

In much the same way we can express the average and instantaneous acceleration of a body as follows.

Using separation of variables on the last equation we can develop an integral relationship between acceleration, velocity, and time.

This says the ending velocity is equal to the beginning velocity plus the integral of the acceleration with respect to time. If the acceleration is constant through the entire time period we get:

Next let’s do the same separation and integration to the position and velocity differential equation.

Now we could proceed and find the relationship between position and constant velocity as we did above, but I think that step is obvious and I leave that to the reader. Here I want to make a substitution and come up with a different relationship. We will substitute the above equation relating velocity and constant acceleration into the last integral as follows.

This last equation can tell us the final position of a body if we know the original position, the original velocity, the constant acceleration rate, and the time of the trip.

Finally we will develop one more kinematic formula. We start again with the differential equation relating velocity and position. Then we will use a chain rule and variable separation with integration to reach our destination.

 

Notice that this equation does not require knowledge about the time. All we need is the original velocity, the constant acceleration rate, and the change in the position to find the final velocity.

I hope you see from this discussion that kinematics is a very simple, fundamental exercise in mechanics. If we boil all of this down, we can solve any problem in general by simply applying the principles of calculus to the  two fundamental differential equations:

to formulate whatever relationship we need.

About jonathanmcgehee

Forensic Engineer, Math Nerd, Husband of @rebekahmcgehee, Father of 4, Follower of Jesus

Posted on March 15, 2011, in Engineering, Engineering Mechanics and tagged , , , , , , , , . Bookmark the permalink. 4 Comments.

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