Free · Linear motion
Free-Fall Arrival Velocity, Drop Distance & Time
Solve the downward displacement, elapsed time and initial velocity of an ideal fall under constant gravity, preserving reverse inputs and unit choices.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
With downward initial velocity 2 m/s and constant acceleration 10 m/s², a 3-second fall covers 51 m below its starting position. Its arrival velocity is 32 m/s downward.
h = v₀t + ½gt²; t = [−v₀ + √(v₀² + 2gh)] / g; v = v₀ + gt
Worked example
Enter these known values and leave the other values blank.
- Constant downward acceleration g
- 10 m/s²
- Signed initial velocity, downward positive
- 2 m/s
- Distance below the starting point h
- 51 m
- Arrival time t
- 3 sec
- Downward arrival velocity v
- 32 m/s
Assumptions and limitations
- Gravity is constant and positive; air resistance and buoyancy are omitted. Enter local gravity when standard gravity is not the intended model.
- Downward is positive. A negative initial velocity represents an upward launch; h is a nonnegative displacement below the starting point, not total path length.
- The square-root branch selects arrival while moving downward. For an upward launch returning to h = 0, it gives the later return time rather than the launch instant.
- Time and drop distance are nonnegative. This is a nonrelativistic model and does not include terminal speed or changing gravity with altitude.
- The gravity default is the conventional standard value, not a measurement of gravity at the user’s location.
Common questions
Can initial velocity point upward?
Yes. Enter it as a negative value. The selected root includes the upward travel before the object returns and reaches the specified point below its start.
Does height mean altitude above sea level?
No. It is the vertical distance below the starting position in this motion model.
References
- OpenStax: constant-acceleration free fall
- NIST: standard acceleration of gravity
- NIST: SI and customary unit definitions
- Calculation definition and unit reference
Bookify makes the metric and imperial gravity defaults represent the same standard 9.80665 m/s², correcting the captured rounded or mistyped imperial preset. Feet per second squared use exactly 0.3048 meters per second squared, replacing a rounded reciprocal factor.