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Explanation of Gravity Velocity Equations for Objects Projected Downward by Ron Kurtus - Succeed in Understanding Physics. Also refer to physical science, acceleration, displacement, time, calculation, square-root, School for Champions. Copyright © Restrictions
Gravity Velocity Equations for Objects Projected Downward
by Ron Kurtus (revised 7 January 2011)
When you throw or project an object downward, it is accelerated until it is released at some initial velocity. If you know this initial velocity, there are simple derived equations that allow you to calculate the velocity when the object reaches a given displacement from the starting point or when it reaches a given elapsed time.
Examples illustrate these equations.
Note: You normally do not need to memorize these equations, but you should know where to find them in order to solve equations.
Questions you may have include:
- How do you find the velocity for a given displacement?
- How do you find the velocity for a given time?
- What are some examples of these equations?
This lesson will answer those questions.
Useful tools: Metric-English Conversion | Scientific Calculator.
Velocity with respect to displacement
The general gravity equation for velocity with respect to displacement is:
v = ±√(2gy + vi2)
where
- ± means plus or minus
- v is the vertical velocity in meters/second (m/s) or feet/second (ft/s)
- √(2gy + vi2) is the square root of the quantity (2gy + vi2)
- y is the vertical displacement in m or ft
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
- vi is the initial vertical velocity of the object
(See Derivation of Displacement-Velocity Gravity Equations for details of the derivation.)
Since v is a downward vector, it has a positive value. Likewise, y and vi are positive numbers. Thus, only the + version of the equation applies:
v = √(2gy + vi2)
Velocity of object projected downward as a function of displacement or time
Velocity with respect to time
The general gravity equation for velocity with respect to time is:
v = gt + vi
where t is the time the object has fallen in seconds (s).
(See the Derivation of Velocity-Time Gravity Equations lesson for details of the derivation.)
This same equation applies for an object projected downward.
Examples
The following examples illustrate applications of the equations.
For a given displacement
Find the velocity of a rock that is thrown down at 2 m/s after it has traveled 2 meters.
Solution
You are given that vi = 2 m/s and y = 2 m. Since vi is in m/s and y is in meters, then g = 9.8 m/s2. The equation to use is:
v = √(2gy + vi2)
Substitute values in the equation:
v = √[2*(9.8 m/s2)*(2 m) + (2 m/s)2]
v = √(39.2 m2/s2 + 4 m2/s2)
v = √(43.2 m2/s2)
v = 6.57 m/s
For a given time
Suppose you throw the object downward at 10 m/s. Find its velocity after 4 seconds.
Solution
You are given that vi = 10 m/s and t = 4 s. Since vi is in m/s, g = 9.8 m/s2. The equation to use is:
v = gt + vi
Substitute values in the equation:
v = (9.8 m/s2)*(4 s) + 10 m/s
v = 39.2 m/s + 10 m/s
v = 49.2 m/s
Summary
You can calculate the velocity when an object that is projected downward reaches a given displacement from the starting point or when it reaches a given elapsed time from the equations:
v = √(2gy + vi2)
v = gt + vi
Help other people learn
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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