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Vectors in Gravity Equations

by Ron Kurtus (revised 10 March 2011)

A vector is a quantity that has magnitude and direction. Vectors are usually designated as arrows pointing in a specific direction, with the length as the magnitude. This provides a way to visualize and to combine entities geometrically.

Using vectors is an effective way to describe the motion of objects in gravity equations, as well as to set a convention for direction in both the vertical and horizontal axes. Vectors are also useful to describe motion at an angle to the force of gravity, since the vector can be broken into its vertical and horizontal components.

If the quantity has only a magnitude and does not have a specific direction, it is a scalar quantity.

Questions you may have include:

This lesson will answer those questions. Useful tool: Units Conversion



Vectors

Certain motion entities not only have magnitude but also have a direction. They can be represented geometrically as arrows or vectors.

Vectors are quantities with magnitude and direction. For example, a truck is traveling north at a velocity of 50 miles per hour. Velocity is a vector with north as the direction and 50 miles per hour as the magnitude.

Vectors used in gravity equations include:

Note: Textbooks often denote vectors as with a special marker over the vectors as an indicator or in boldfaced. In our material, vectors, scalars and equations are boldfaced to distinguish them from the other text. If an item is a vector, it will be noted as such.

Gravity vector

Gravity vector

Coordinate system

When dealing with vectors and gravity equations, the x-y or Cartesian coordinate system is used, with the y-axis as vertical and the x-axis as horizontal. The zero-point of the axis is the starting point of the object's motion.

Direction convention

The convention for direction that we use is that vectors toward the ground are positive and those upward are negative. This essentially is inverting or flipping the usual x-y coordinate system, such that the −y-axis is up instead of downward.

(See Convention for Direction in Gravity Equations for more information.)

Magnitude of vectors

The magnitude of a vector is always a positive number, no matter what the direction of the vector. It is an indication of the length of the geometric representation, as well as a multiplier when comparing two vectors.

Vector components

Vectors at an angle to the ground or horizontal plane can be broken into vertical and horizontal components. This includes velocity and displacement. Obviously, gravity is only in the vertical direction.

Angles

The convention for angles is that they are measured clockwise from the horizontal or x-axis. An angle measured counterclockwise from the x-axis results in a negative angle or 360° minus the angle.

Geometric visualization

Consider the velocity at an upward angle from the ground. It would be represented by a negative vector, −v. The angle from the horizontal would be counterclockwise or a negative angle, θ (Greek letter theta).

Perpendicular components of velocity at a negative angle

Perpendicular components of velocity at a negative angle

The velocity vector can be broken into its perpendicular components, where:

−vy = −v*sin(−θ)

and

vx = −v*cos(−θ)

Also, according to the Pythagorean Theorem:

v2 = vx2 + vy2

Scalars

If the quantity only has a magnitude and does not have a direction, it is a scalar quantity. Scalar quantities can only be positive.

Scalar quantities used in gravity equations are:

Speed versus velocity

Speed is how fast an object is going irrespective of its path. Velocity is a vector that is the speed in a specific direction.

Distance versus displacement

While displacement is a vector that shows how far an object moves in some direction, distance is the total of the path taken between two points.

Vectors as scalars

The absolute or positive value of a vector is its magnitude or scalar quantity. For example:

|v| = s

|y| = d

where

Likewise:

|−v| = s

|−y| = d

Summary

A vector is a quantity that has magnitude and direction. Using vectors is an effective way to describe the motion of objects in gravity equations. They are also are also useful to describe motion at an angle to the force of gravity, since the vector can be broken into its vertical and horizontal components.

If the quantity has only a magnitude and does not have a specific direction, it is a scalar quantity. A scalar quantity is also the absolute value of its vector.


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Resources and references

Ron Kurtus' Credentials

Websites

Gravity Resources

Vectors - Fundamentals and Operations - Physics Classroom

Basic Vector Operations - HyperPhysics

Vectors and Direction - Physics Classroom

Books

Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics


Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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