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Explanation of Derivation of Velocity-Time Gravity Equations - Succeed in Understanding Physics. Also refer to physical science, falling objects, Calculus, acceleration, velocity, integral, derivative, integrate, distance, time, relationships, Ron Kurtus, School for Champions. Copyright © Restrictions
Derivation of Velocity-Time Gravity Equations
by Ron Kurtus (revised 2 July 2010)
The basis for the derivations of the velocity-time gravity equations starts with the assumption that the acceleration due to gravity is a constant value.
Since acceleration is also the change in velocity for an increment of time, you use Calculus to integrate that change to get the velocity for a given elapsed time.
From the velocity equation, you can then determine the equation for the time it takes for the object to reach a given velocity from the starting point.
Questions you may have include:
- What is the basis for the derivations?
- What is the velocity for a given time equation?
- What is the time for a given velocity equation?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Basis for velocity-time derivations
The derivations start with the assumption that the acceleration due to gravity, g, is a constant for distances relatively close to Earth.
Acceleration is also the incremental change in velocity with respect to time:
a = dv/dt
where
- a is the acceleration
- dv is the first derivative of velocity v (a small change in velocity)
- dt is the first derivative of time t (a small time increment)
Since g is acceleration:
a = g
and
dv/dt = g
Multiply both sides of the equation by dt to get:
dv = g*dt
By using Calculus to integrate this equation, you can get the equations for velocity and time.
Velocity-time relationship
Derivation of velocity for a given time
Integrate dv = g*dt on both sides of the equal sign.
First, integrate dv over the interval from vi to v:
∫dv = v − vi
where
- ∫ is the integral sign, as used in Calculus
- vi is the initial velocity of the object
Then, integrate g*dt over the time interval from 0 to t:
∫g*dt = gt
Thus, the equation for the velocity of a falling object with respect to time and an initial velocity of vi is:
v − vi = gt
v = gt + vi
Derivation of time for a given velocity
The time it takes to reach a given velocity is obtained by rearranging the equation v = gt + vi and solving for t:
v = gt + vi
v − vi = gt
t = (v − vi)/g
Summary
Starting with the fact that the acceleration due to gravity, g, is considered a constant and knowing that acceleration is the change in velocity for a change in time, you can derive the gravity equations for the velocity with respect to time. You can then determine the equation for the time to reach a given velocity.
The derived equations are:
v = gt + vi
t = (v − vi)/g
See the Side Menu for more Gravity and Gravitation topics
Know where equations come from
Resources
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Gravity and Gravitation Resources
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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Derivation of Velocity-Time Gravity Equations