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Explanation of Derivation of Velocity-Time Gravity Equations by Ron Kurtus - Succeed in Understanding Physics. Also refer to physical science, falling objects, Calculus, acceleration, velocity, integral, derivative, integrate, displacement, time, relationships, School for Champions. Copyright © Restrictions
Derivation of Velocity-Time Gravity Equations
by Ron Kurtus (revised 7 January 2011)
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.
Useful tool: Metric-English Conversion
Basis for velocity-time derivations
The derivations start with the assumption that the acceleration due to gravity g is a constant for displacements 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)
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.)
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 v = vi to v = 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
Note: The initial velocity is the velocity at which the object is released after being accelerated from zero velocity. Initial velocity does not occur instantaneously.
Then, integrate g*dt over the time interval from t = 0 to t = t:
∫g*dt = gt − 0
The result of the two integrations is:
v − vi = gt
Thus, the general gravity equation for velocity with respect to time is:
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 − 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
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
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Derivation of Velocity-Time Gravity Equations