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Explanation of Derivation of Displacement-Velocity Gravity Equations by Ron Kurtus - Succeed in Understanding Physics. Also refer to physical science, acceleration, velocity, displacement, time, relationships, square root, School for Champions. Copyright © Restrictions
Derivation of Displacement-Velocity Gravity Equations
by Ron Kurtus (revised 7 January 2011)
The basis for the derivations of the displacement-velocity gravity equations starts with the equations t = (v − vi)/g and y = gt2/2 + vit, as determined in the Derivation of Velocity-Time Gravity Equations and Derivation of Displacement-Time Gravity Equations lessons.
Note: Displacement is a vector quantity denoting the change in position in a given direction. Velocity is a vector indicating the rate of change of position in a given direction.
By substituting the time relationship in the displacement equation, you can determine the displacement with respect to velocity. From the derived displacement equation, you can then determine the equation for the velocity when the object reaches 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 velocity equation?
- What is the velocity for a given displacement equation?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Basis for displacement-velocity derivations
To determine the displacement from the starting point at a given velocity, start with the equations:
t = (v − vi)/g
(Obtained from Derivation of Velocity-Time Gravity Equations)
and
y = gt2/2 + vit
(Obtained from Derivation of Displacement-Time Gravity Equations)
where
- t is the time in seconds (s)
- v is the vertical velocity in m/s or ft/s
- vi is the initial vertical velocity in m/s or ft/s
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
- y is the vertical displacement from the starting point in m or ft
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.
By substituting the equation for t in the equation for y, you can get the displacement with respect to velocity. Then by solving that equation for v, you get the velocity with respect to displacement equation.
Displacement-velocity relationship
Derivation of displacement for a given velocity
To derive the displacement equation, you can start with the time equation:
t = (v − vi)/g
Square both sides of the equation:
t2 = (v − vi)2/g2
t2 = (v2 − 2vvi + vi2)/g2
Consider the displacement equation:
y = gt2/2 + vit
Substitute for t2 and t from the above equations:
y = g (v2 − 2vvi + vi2)/2g2 + vi(v − vi)/g
Multiply vi(v − vi)/g by 2/2 and combine like terms:
y = (v2 − 2vvi + vi2)/2g + 2(vvi − vi2)/2g
y = (v2 − 2vvi + vi2 + 2vvi − 2vi2)/2g
The resulting general gravity equation for displacement with respect to velocity is:
y = (v2 − vi2)/2g
Derivation of velocity for a given displacement
To get the velocity for a given displacement, multiply both sides of
y = (v2 − vi2)/2g by 2g and solve for v:
2gy = (v2 − vi2)
v2 = 2gy + vi2
Take the square root of both sides of the equation to get the general gravity equation for velocity with respect to displacement:
v = ±√(2gy + vi2)
where
- ± means plus or minus
- √(2gy + vi2) is the square root of the quantity (2gy + vi2)
Summary
The gravity equation for the displacement an object travels from the starting point until it reaches a given velocity can be derived from the equations t = (v − vi)/g and y = gt2/2 + vit. This leads to the equation for the velocity when the object reaches a given displacement from the starting point.
The derived equations are:
y = (v2 − vi2)/2g
v = ±√(2gy + vi2)
Be observant and curious
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 Displacement-Velocity Gravity Equations