Monthly Archives: March 2011

The Hyperbola

Ready to take a look at the third conic section? (The first two we looked at were the parabola and the ellipse). The hyperbola is created when the plane passes through both nappes of the right circular conical surface at an angle with the surface’s longitudinal axis that is less than the angle made with the axis by the generating straight line on the surface.

The hyperbola is found in many natural phenomena including an open orbit of celestial bodies or simply the curvature of light around a large gravitational body like a star. It is also the path of subatomic particles repulsed by the nucleus. In more everyday terms, the tip of a shadow cast by the sun traces out a hyperbola on the ground as the day progresses. The mathematics of hyperbolas is also used in navigation and global positioning systems.

The definition of a hyperbola that we will use to develop an algebraic expression is similar to that of the ellipse. A hyperbola is all the points in a plane the difference of whose distances from the two foci is a (positive) constant. The ellipse was simply the sum instead of the difference.

We begin with the hyperbola centered on the origin in a Cartesian-coordinate plane. As with the ellipse we use the foci, (c,0) and (-c,0). The distance from each focus to a general point (x,y) is given by the following two expressions using the two-dimensional distance formula:

Now using the above definition we will set the difference between these two to a constant number, 2a, just as we did with the ellipse. One important thing to note is since as we move about the plane the difference between these may change from negative to positive or positive to negative depending on which distance is longer. We will use the absolute value to insure we get the magnitude of the difference in distance.

As with the ellipse, using some elementary algebra we arrive at:

Now “a” represents the x-intercepts or the vertices. (Let y=0 and multiply both sides by “a” squared). With the ellipse, “a” was always greater than “c” so we factored out a negative 1 from the denominator of the second term and changed the sign between the two terms to a positive. Then we used the substitution:

In the present case, “a” will always be less than “c” so we use the substitution:

This yields the final expression:

Notice here that if we try to locate the y-intercepts by setting x=0, we get imaginary results. (Square root of negative “b” squared). Therefore this is the expression for the hyperbola with vertices on the x-axis. The expression for the hyperbola with vertices on the y-axis is:

Just as with the parabola we can use the same substitutions for translation and axes and rotation of axes to develop expressions for more general hyperbolas.

The axis that runs through the vertices is called the transverse axis. The axis that is orthogonal to the transverse axis is the conjugate axis. A hyperbola is often drawn by constructing a rectangle that is 2a by 2b in dimension with the vertices on the 2a length sides. Then diagonals are drawn through the rectangle and used as asymptotes for the hyperbolic curve. The equation for these asymptotes can be derived by solving the main equations for y.

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.