What is a Parabola? A 5 Minute Introduction

Definition/Summary

A parabola has many definitions, a classical one being, “A Parabola is the locus of all points equidistant from a given point (called the focus) and a line (called the directrix)”.

A parabola is a frequently encountered trajectory in kinematics and mechanical physics. Also, the plot of a Quadratic Equation in one variable is a parabola.

Glossary:

Vertex: There exists a point on the parabola, from where if a tangent to the parabola is drawn, the tangent is parallel to the directrix. This point is known as the ‘vertex’ of the parabola.

Latus Rectum: A chord of a parabola, which passes through the focus and is parallel to the directrix is known as the latus rectum of the parabola.

Axis: the line through the focus and the vertex, and therefore perpendicular to the directrix and the latus rectum. It is an axis of reflectional symmetry of the parabola and corresponds to the major axis of an ellipse.

Equations

The standard form of a parabola, with the directrix parallel to the y-axis and the origin as the vertex:

$$y^2 = 4ax$$

In this parabola, the equation of the directrix is given as:

$$x = -a$$

and the focus, S is given as:

$$S(x, y) \equiv (a, 0)$$

Such a parabola is symmetric about the x-axis.

A parabola with vertex at (f, g) is given as:

$$(y – g)^2 = 4a(x – f)$$

Parametric form of a point P on a parabola is given as:

$$P(x, y) \equiv (at^2, 2at)$$

and also,

$$P(x, y) \equiv (\frac{a}{m^2}, \frac{2a}{m})$$

where, $m$ is the slope of the tangent at that point.

The equation of a tangent and the normal at a point $(at^2, 2at)$ is given as:

$$ty = x + at^2$$

$$y + tx = 2at + at^2$$

Extended explanation

Alternative definitions for a parabola:

i] A parabola is a conic section with eccentricity $e = 1$.

ii] cited from Wikipedia:

[quote]In mathematics, the parabola (pronounced /pəˈræbələ/, from the Greek παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface.

[/quote]

iii] A Parabola is the limiting case of an ellipse with one focus at infinity.

Parabolas and gravity:

A projectile in a uniform gravitational field follows a parabola (unless moving vertically). A ball thrown into the air at an angle is an example of projectile motion. The ball follows a parabolic path.

The Earth’s gravitational field is of course spherical, not uniform, and so a projectile follows an ellipse, but the difference is so small that, over short distances, the ellipse is indistinguishable from a parabola.

A projectile moving at escape velocity (in any direction) in a Newtonian spherical gravitational field follows a parabola (unless moving vertically). A projectile moving more slowly follows an ellipse, and a projectile moving faster follows a hyperbola.

NOTE: A hanging chain, or hanging string having mass, does not trace a parabolic arc, as it is commonly assumed. The shape it traces is a catenary. The main chain of a suspension bridge, however, which supports a mass uniformly distributed horizontally, traces a parabolic arc.

Common encounters with a paraboloid:

A paraboloid is a surface obtained by rotating a parabola about its axis.

i] The surface of a fluid in a rotating vessel is (part of) a paraboloid.

ii] Most reflecting telescopes have paraboloid mirrors (“parabolic mirrors”) because light from a very distant object (effectively “at infinity”) on the axis of the telescope is focused onto … the focus!

See also https://www.physicsforums.com/threads/what-is-the-equation-of-a-circle.762871/ and https://www.physicsforums.com/threads/what-is-a-conic.762859/

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