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Derivation of Distance-Time Gravity Equations
by Ron Kurtus (revised 2 July 2010)
The basis for the derivation of the distance-time gravity equations starts with the equation v = gt + vi that was determined in the Derivation of Velocity-Time Gravity Equations lesson.
Since velocity is the change in distance over an increment in time, you use Calculus to integrate that change and get the distance for a given elapsed time.
From that distance equation, you can then determine the equation for the time it takes for the object to reach a given distance from the starting point.
Questions you may have include:
- What is the basis for the derivations?
- What is the distance for a given time equation?
- What is the time for a given distance 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 distance-time derivations
To determine the distance from the starting point for a given time, start with the equation:
v = gt + vi
(Obtained from Derivation of Velocity-Time Gravity Equations)
where
- v is the vertical velocity 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
Velocity is also the incremental change in distance with respect to time:
v = dy/dt
where
- dy is the first derivative of vertical distance y
- dt is the first derivative of time t
By substituting combining these two equations and integrating, you can derive the distance with respect to time. Then you can rearrange the equation and solve for t to get the time with respect to distance.
Distance-time relationship
Derivation of distance with respect to time
To obtain the distance with respect to time, substitute for v in v = gt + vi:
dy/dt = gt + vi
Multiply both sides of the equation by dt:
dy = gt*dt + vi*dt
Integrate dy over the interval from 0 to y:
∫ dy = y
where
- ∫ is the integral sign between the two limits
- y is the vertical distance from the starting point
Integrate gt*dt over the interval from 0 to t:
∫gt*dt = gt2/2
Integrate vi*dt over the interval from 0 to t:
∫vi*dt = vit
The result is:
y = gt2/2 + vit
Derivation of time with respect to distance
You can find the time it takes for an object to travel a given distance from the starting point by rearranging y = gt2/2 + vit and solving the quadratic equation for t:
y = gt2/2 + vit
Subtract y from both sides of the equation and multiply both sides by 2.
gt2 + 2vit − 2y = 0
Solve the quadratic equation for t:
(See Using the Quadratic Equation Formula in our Algebra section for more information.)
−2vi ± √(4vi2 + 8gy)
t = ________________
2g
Since that is difficult to write on a web page and may not display well on some browser configurations, we will use the following version of the equation, which is also a more compact form:
t = [ −2vi ± √(4vi2 + 8gy) ]/2g
Remove the square root of 4 from inside the square root or radical sign:
t = [ −2vi ± 2√(vi2 + 2gy) ]/2g
t = [ −vi ± √(vi2 + 2gy) ]/g
where
- ± means plus or minus
- √(vi2 + 2gy) is the square root of the quantity (vi2 + 2yg)
The plus-or-minus sign means that in some situations, there can be two values for t for a given value of y.
Summary
The basis for the derivation of the distance-time gravity equations starts with the equation v = gt + vi. Since velocity is the change in distance over an increment in time, you integrate that change and get the distance for a given elapsed time.
From that distance equation, you can then determine the equation for the time it takes for the object to reach a given distance from the starting point.
The derived equations are:
y = gt2/2 + vit
t = [−vi ± √(vi2 + 2gy)]/g
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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 Distance-Time Gravity Equations