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Derivation of Velocity-Time Gravity Equations
by Ron Kurtus (30 August 2009)
You can derive the gravity equations for the velocity of an object that is dropped, thrown downward or projected upward after a given time, as well as the time for a given velocity.
The derivation starts with the fact that the force of gravity is F = mg, where g is the acceleration due to gravity and is a constant value. Acceleration is the change in velocity for a change in time, so you can use Calculus to integrate that change to get the velocity for a given elapsed time. You can also then determine the equation for the time to reach a given velocity.
The initial velocity of the object determines the starting direction. If the object is thrown upward, there are two possible velocities—one for upward motion and another as the object falls downward.
Advanced Physics and Physical Science students are often required to derive equations. Beginning students who have not yet taken Calculus can skip or skim this lesson.
(See Overview of Derivation of Gravity Equations for a summary of the derivations.)
Questions you may have include:
- What is the velocity for a given time equation?
- What is the time for a given velocity equation?
- What is the effect of the initial velocity?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Velocity for a given time
The velocity of a falling object for a given time is found by starting with the definition of acceleration as the change in velocity over a period of time:
a = dv/dt
where
- a is 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 the acceleration due to the force of gravity, g, is constant for distances relatively close to Earth, g = a, and thus g = dv/dt.
Multiply both sides of the equation by dt to get:
dv = g*dt.
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
Integrate g*dt over the 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
Positive and negative values
Since the direction of the force of gravity is downward, we consider velocity in that direction as positive and those upward and away from the ground as negative.
Note: Some text books use a different convention, saying the up is positive. However, that does not seem to follow good logic, since the positive force is toward te ground.
If the initial velocity is upward, vi is a negative number. When the object is moving upward, v is also negative. Once the object is moving downward, v becomes a positive number.
Obviously, t can only be positive.
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
Effect of initial velocity
The direction of the initial velocity affects the direction of the velocity after a period of time.
Projected downward
Since the direction toward the ground is considered positive, the initial velocity is positive when the object is thrown downward and its value adds to the object's velocity.
Thrown upward
If the object is thrown upward, the initial velocity is negative. This results in the velocity first being negative, slowing to zero at some maximum height and then becoming positive and continuing downward.
Example
If g = 32 ft/s2 and vi = −64 ft/s, then applying the equation v = gt + vi, the direction and velocity of the object at various times are:
t = 1 s v = −32 ft/s Moving upward t = 2 s v = 0 At maximum height t = 3 s v = +32 ft/s Moving downward,
above starting pointt = 5 s v = +96 ft/s Moving downward,
below starting point
Initial velocity is zero
In the case where the object is just dropped, vi = 0, and the equations simply become:
v = gt
t = v/g
Summary
Starting with the fact that the acceleration due to gravity, g, is considered a constant, you can derive the gravity equations for the velocity of an object after a given time, as well as the time for a given velocity. Since acceleration is the change in velocity for a change in time, you integrate that change to get the velocity for a given elapsed time. You can also 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 Gravitation and Gravity 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
Gravitation and Gravity 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