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Convention for Direction in Gravity Equations

by Ron Kurtus (revised 15 March 2011)

Since some gravity equations result in an object can changing directions, there needs to be a convention as to which direction is positive and which is negative.

Due to the fact that the force of gravity is downward, it would only seem logical to define downward as a positive direction in gravity equations. This is essentially inverting the Cartesian coordinate system. Using vectors is an effective way to describe the initial and resulting motion of the objects and to set a convention for direction.

We consider the displacement direction as positive below the starting point and negative above. Velocity is positive in a downward direction and negative when going upward. For horizontal motion, right is positive and left is negative.

Questions you may have include:

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



Gravity convention

The acceleration due to gravity, g, is in a direction toward the ground. The same is true of the force of gravity, F = mg.

Both g and F can be considered vectors, which are geometric representations indicating both magnitude and direction. Vectors are typically drawn as arrows, with their length proportional to their magnitude. The magnitude of g is 9.8 m/s2 or 32 ft/s2.

Since both the acceleration due to gravity and the force of gravity are downward, we use the convention for our gravity equations that downward vectors are positive (+). Likewise, upward vectors are negative (−).

In essence, this is flipping the x-y or Cartesian coordinate system, such that the +y-axis points down and the −y-axis points up.

This convention affects the other vectors used in gravity equations, which include:

Displacement convention

Displacement is the amount of movement or change in position from the starting point in a given direction. It is a vector, with a direction of either downward, upward or at an angle with the vertical line.

Note: Displacement is often confused with distance. Displacement concerns a direct path from one point to another and is a vector. Distance is a scalar quantity where the path does not matter.

Magnitude

The magnitude of displacement is the separation between the starting and end points along the vector line.

Distance can also be considered as the magnitude of displacement, provided it is along the vector line.

Vertical displacement

Vertical displacement can be represented as a vector that is positive below the starting point and in the direction of gravity. It is negative above the starting point.

Note that when an object is projected upward, reaches its maximum displacement and starts moving downward, the displacement is negative above the starting point and positive below the starting point. Displacement is measured from the starting point.

Positive and negative displacement vectors

Positive and negative displacement vectors

Horizontal displacement

Displacement vectors in a horizontal direction are designated as positive toward the right direction and negative toward the left. Vectors at an angle can be broken into their components on the x-y axes.

Velocity convention

Velocity is a vector that is the change in displacement with respect to time in a specific direction. It is measured from a given displacement.

Velocities in the same direction as gravity are positive and velocities in the opposite direction are negative vectors.

Positive and negative velocity vectors

Positive and negative velocity vectors

Magnitude

The magnitude of a velocity vector is the speed in the given direction. It is also the absolute or positive value of the velocity.

Velocity at a negative displacement

It is possible for a velocity at a negative displacement to be in the direction of gravity and be a positive vector. This is seen in the case of throwing a ball upward and having it fall toward the ground.

Positive velocity from negative displacement

Positive velocity from negative displacement

Velocity in horizontal direction

Velocity vectors in a horizontal direction are designated as positive toward the right direction and negative toward the left. Vectors at an angle can be broken into their components on the x-y axis.

Summary

Gravity equations require a convention, stating which direction is positive and which is negative. Using vectors is an effective way to describe the initial and resulting motion of the objects and to set a convention for direction.

We consider the direction of gravity as positive and the upward direction as negative. Displacement is positive below the starting point and negative above. Velocity is positive in a downward direction and negative when going upward. For horizontal motion, right is positive and left is negative.


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Resources and references

Ron Kurtus' Credentials

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Vectors and Direction - Physics Classroom

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Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics


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