Using Vectors in Gravity Equations
by Ron Kurtus (revised 6 February 2018)
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.
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.
If the quantity has only a magnitude and does not have a specific direction, it is a scalar quantity.
Questions you may have include:
- What are the gravity vectors?
- How are vectors at an angle handled?
- What are scalar quantities in gravity equations?
This lesson will answer those questions. Useful tool: Units Conversion
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:
- Acceleration due to gravity: g
- Force of gravity: F
- Velocity: v
- Displacement: y and x
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.
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.
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 angle 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.
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.
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
The velocity vector can be broken into its perpendicular components, where:
−vy = −v*sin(−θ)
vx = −v*cos(−θ)
Also, according to the Pythagorean Theorem:
v2 = vx2 + vy2
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:
- Mass: m
- Time: t
- Speed: s
- Distance: d
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
- |v| is the absolute or positive value of the velocity, independent of the direction
- |y| is the absolute value of the displacement, independent of direction
|−v| = s
|−y| = d
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.
Consider yourself a lucky person
Resources and references
Vectors - Fundamentals and Operations - Physics Classroom
Basic Vector Operations - HyperPhysics
Vectors and Direction - Physics Classroom
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Using Vectors in Gravity Equations