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Gravity Time Equations for Objects Projected Downward
by Ron Kurtus (28 August 2009)
If you know the initial velocity that an object is thrown or projected downward, you can calculate the time it takes for it to reach a given velocity or reach a given distance from the starting point. Some examples illustrate these equations.
The equations were determined from the Derivation of Gravity Equations and are summarized in Overview of Gravity Equations for Objects Projected Downward.
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 time for a given velocity?
- How do you find the time for a given distance?
- What are some examples of these equations?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Time to reach velocity
The equation for the time it takes an object that is thrown or projected downward to reach a given velocity is:
t = (v − vi)/g
where
- t is the time in seconds (s)
- v is the velocity of the falling object in feet/second (ft/s) or meters/second (m/s)
- vi is the initial velocity the object has been thrown downward in ft/s or m/s
- g is the acceleration due to gravity; g = 32 ft/s2 or 9.8 m/s2
Time to reach distance
The equation for the time it takes an object projected downward at some initial velocity to reach a given distance is:
−vi ± √(vi2 + 2gx)
t = ________________
g
Or, it can be written in a more compact form:
t = [−vi + √(vi2 + 2gx)]/g
where
- x is the distance traveled from the starting point in feet (ft) or meters (m)
- √(vi2+ 2gx) is the square root of the quantity (vi2+ 2gx)
Elapsed time of an object thrown downward as a function of
initial velocity and velocity or distance
Examples
The following examples illustrate applications of the equations.
For a given velocity
If you throw a ball downward from a tall building at 5 ft/s, find the time it takes for the ball to reach a velocity of 101 ft/s.
Solution
You are given that vi = +5 ft/s and v = 101 ft/s. Since vi and v are
in ft/s,
g = 32 ft/s2. The equation to use is:
t = (v − vi)/g
Substitute values in the equation:
t = (101 ft/s − 5 ft/s)/(32 ft/s2)
t = (96 ft/s)/(32 ft/s2)
t = 3 s
For a given distance
If you throw an object downward from a high building at 5 m/s, find the time it takes to fall 50 m.
Solution
You are given that vi = +5 m/s and x = 50 m. Since vi in m/s and x is in m,
g = 9.8 m/s2. The equation to use is:
t = [−vi + √(vi2 + 2gx)]/g
Substitute values in the equation:
t = [−5 m/s + √{(25 m/s)² + 2*(9.8 m/s²)*(50 m)}]/(9.8 m/s²)
t = [−5 m/s + √(625 m²/s² + 980 m²/s²)]/(9.8 m/s²)
t = [−5 m/s + √(1605 m²/s²)]/(9.8 m/s²)
t = [−5 m/s + 40.1 m/s]/(9.8 m/s²)
t = (35.1 m/s)/(9.8 m/s²)
t = 3.58 s
(Whew!)
Summary
You can calculate the time it takes an object that is projected downward to reach a given velocity or reach a given distance from the starting point from the equations:
t = (v − vi)/g
t = [−vi + √(vi2 + 2gx)]/g
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Resources
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Gravitation and Gravity 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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Gravity Time Equations for Objects Projected Downward