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Derivation of Velocity-Time Gravity Equations
by Ron Kurtus (revised 10 August 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.
The derived equations are affected by the initial velocity of the object. This is important in later applications of the equations.
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 the acceleration due to gravity:
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
- v is the vertical velocity of the object
- vi is the initial vertical velocity of the object
Then, integrate g*dt over the time interval from 0 to t:
∫g*dt = gt − 0
The result of the integrations is:
v − vi = gt
The general gravity equation for velocity with respect to time is:
v = gt + vi
However, the value of the initial velocity affects the equation.
vi = 0
When the object is simply dropped, the initial velocity is zero (vi = 0) and the equation becomes:
v = gt
vi > 0
When the object is thrown downward, the initial velocity is greater than zero
(vi > 0) and the velocity is also positive.
vi < 0
When the object is projected upward, the initial velocity is less than zero (vi < 0) and v is negative on the way up, zero at the maximum height and positive on the way down.
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
Again, the value of the initial velocity affects the equation.
vi = 0
When the object is dropped, the initial velocity is zero (vi = 0) and the equation becomes:
t = v/g
vi > 0
Since t cannot be negative, v must be greater than or equal to vi when the object is thrown downward.
vi < 0
When the object is projected upward, both v and vi are negative on the way up. At the maximum height, v = 0 and the equation becomes:
tm = − vi/g
where
- tm is the time to reach the maximum height
- − vi is a positive number
Then, on the way down, v is a postive number.
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
tm = − vi/g (at maximum height when projected upward)
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