SfC Home > Physical Science > Physics > Gravity >
Explanation of Overview of Gravity Equation Derivations by Ron Kurtus - Succeed in Understanding Physics. Also refer to physical science, falling objects, Calculus, acceleration, velocity, integral, derivative, integrate, displacement, distance, time, relationships, School for Champions. Copyright © Restrictions
Overview of Gravity Equation Derivations
by Ron Kurtus (revised 7 January 2011)
Starting with the assumption that the acceleration due to gravity is a constant value, you can derive equations that define the relationships between velocity, displacement and time for an object moving under the influence of gravity. An initial velocity factor is included in these equations.
You use basic Calculus to determine the equations for the relationship between velocity and time for an object that is dropped, thrown downward or projected upward. From the velocity equation, the displacement-time relationship is derived. Then the velocity-displacement equations are obtained from the previous two derivations.
Questions you may have include:
- What are the velocity and time relationships?
- What are the displacement and time relationships?
- What are the velocity and displacement relationships?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Overview of velocity-time relationships
The velocity and time relationships as a result of the force of gravity are based on the fact that the acceleration due to gravity, g, is a constant value.
Since acceleration is the change in velocity with respect to time, the equation for the acceleration due to gravity is:
dv/dt = g
where
- dv is the derivative or small change in velocity
- dt is the derivative or increment of time
Note: Vectors have magnitude and direction and are indicated in boldface. Scalars have only magnitude and are in regular text.
(See Vectors in Gravity Equations for more information.)
Using Calculus, you integrate and derive the relationship between velocity and time for an object under the influence of gravity:
v = gt + vi
and
t = (v − vi)/g
where
- v is the vertical velocity of the object in m/s or ft/s
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
- t is the time in seconds (s)
- vi is the initial vertical velocity in m/s or ft/s
(See Derivation of Velocity-Time Gravity Equations for details of the derivations.)
Overview of displacement-time relationships
The displacement a moving object travels in a given time is found by knowing that velocity is the change in displacement with respect to time:
v = dy/dt
Note: Displacement is a vector quantity denoting the change in position in a given direction, whereas distance is a scalar quantity that denotes the total change in position, independent of the path taken.
Substituting for v in the equation v = gt + vi and integrating, you get:
y = gt2/2 + vit
Rearranging y = gt2/2 + vit and solving the quadratic equation for t gives you:
t = [−vi ± √(vi2+ 2gy)]/g
This equation can create some confusion because of the plus-or-minus sign. If the object is thrown downward, the plus (+) sign is used. If the object is thrown upward, the sign depends on the object's position with respect to the starting point.
(See Derivation of Displacement-Time Gravity Equations for details of the derivations.)
Overview of velocity-displacement relationships
To determine the displacement required to reach a given velocity, start with the equations t = (v − vi)/g and y = gt2/2 + vit from the previous derivations to get:
y = (v2 − vi2)/2g
Solving for v, you get:
v = ±√(2gy + vi2) (See Derivation of Velocity-Displacement Gravity Equations for details of the derivations.)
Summary
The relationships between the velocity of an object under the influence of gravity, the displacement and the time it takes to fall start with a basic equation a = g. You use calculus to integrate equations, use algebra for substitutions and perform other operations to get the results.
Know where equations come from
Resources and references
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
What do you think?
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.
Share link
Click on a button to share the link for this page:
Or use our form to send this link to yourself or a friend.
Students and researchers
The Web address of this page is:
www.school-for-champions.com/science/
gravity_derivations.htm
Please include it as a link on your website or as a reference in your report, document, or thesis.
Where are you now?
Overview of Gravity Equation Derivations