SfC Home > Physics > Gravity >
Derivation of Displacement-Time Gravity Equations
by Ron Kurtus
You can derive the gravity equations for the relationship between displacement and time of an object moving in the vertical direction, starting with the derived velocity-time equations.
Note: Displacement is a vector quantity denoting the change in position in a given direction.
(See Derivation of Velocity-Time Gravity Equations lesson.
Since velocity is the change in displacement over an increment in time, you use Calculus to integrate that change and get the displacement for a given elapsed time. From that displacement equation, you can then determine the equation for the time it takes for the object to reach a given displacement 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 displacement for a given time equation?
- What is the time for a given displacement equation?
This lesson will answer those questions. Useful tool: Units Conversion
Basis for displacement-time derivations
To determine the displacement 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
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.
Velocity is also the incremental change in displacement with respect to time:
v = dy/dt
where
- dy is the first derivative of vertical displacement y
- dt is the first derivative of time t
By substituting combining these two equations and integrating, you can derive the displacement with respect to time. Then you can rearrange the equation and solve for t to get the time with respect to displacement.
Displacement-time relationship
Derivation of displacement with respect to time
To obtain the displacement 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 y = 0 to y = y:
∫dy = y − 0
where
- ∫ is the integral sign between the two limits
- y is the vertical displacement from the starting point
Integrate gt*dt over the interval from t = 0 to t = t:
∫gt*dt = gt2/2 − 0
Integrate vi*dt over the interval from t = 0 to t = t:
∫vi*dt = vit − 0
The result of the integrations is the general gravity equation for the displacement with respect to time:
y = gt2/2 + vit
Derivation of time with respect to displacement
You can find the time it takes for an object to travel a given displacement from the starting point by solving the following quadratic equation for t:
y = gt2/2 + vit
Rearrange the equation by subtracting y from both sides of the equation and multiplying both sides by 2.
gt2 + 2vit − 2y = 0
Solve the quadratic equation for t:
t = [ −2vi ± √(4vi2 + 8gy) ]/2g
(See Using the Quadratic Equation Formula in our Algebra section for more information.)
Remove the square root of 4 from inside the square root or radical sign:
t = [ −2vi ± 2√(vi2 + 2gy) ]/2g
The resulting general gravity equation for time with respect to displacement is:
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 displacement-time gravity equations starts with the equation v = gt + vi. Since velocity is the change in displacement over an increment in time, you integrate that change and get the displacement for a given elapsed time.
From that displacement equation, you can then determine the equation for the time it takes for the object to reach a given displacement from the starting point.
The derived equations are:
y = gt2/2 + vit
t = [−vi ± √(vi2 + 2gy)]/g
Find solutions to your problems by being clever
Resources and references
Websites
Falling Bodies - Physics Hypertextbook
Equations for a falling body - Wikipedia
Gravity Calculations - Earth - Calculator
Books
(Notice: The School for Champions may earn commissions from book purchases)
Top-rated books on Simple Gravity Science
Top-rated books on Advanced Gravity Physics
Questions and comments
Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.
Share this page
Click on a button to bookmark or share this page through Twitter, Facebook, email, or other services:
Students and researchers
The Web address of this page is:
www.school-for-champions.com/science/
gravity_derivations_displacement_time.htm
Please include it as a link on your website or as a reference in your report, document, or thesis.
Where are you now?
Derivation of Displacement-Time Gravity Equations