BIZOMAT MAKEOS · Mechanics
Uniformly Accelerated Motion and Falling Bodies
How fast and how far does this body travel under constant acceleration, or how fast does it hit the ground?
Final velocity and distance from initial velocity, acceleration and time; the acceleration and time needed to change speed over a distance; and the impact speed and time of a body falling from a height.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Known values | choice | Choose which values you know. |
| Initial velocity | m/s | Velocity at the start (downward for a fall). |
| Acceleration (negative to decelerate) | m/s² | Constant acceleration; enter a negative value for braking. |
| Time | s | Duration of the acceleration. |
| Final velocity (distance mode) | m/s | Velocity at the end of the distance. |
| Distance (distance mode) | m | Distance over which the speed changes. |
| Fall height (fall mode) | m | Height of the fall. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Known values
- What it is
- Which set of known values the tool starts from: a speed change over a time, a speed change over a distance, or a body dropped from a height. Only the inputs named for that case are used.
- Units and format
- Choose one of 3 options from the list: Initial speed, acceleration and time; Initial and final speed over a distance; Fall from a height.
- Example
- The form opens with “Initial speed, acceleration and time” selected as a worked example. Choose the option that matches your case.
- Get it right
- Inputs marked for another case are ignored, so a value left in them does not change the result.
What this means: Initial velocity
- What it is
- The speed of the body at the instant the motion being studied begins; zero if it starts from rest. In fall mode it is the downward speed at the moment of release.
- Units and format
- Enter a number in m/s: metres per second. The form accepts 0 m/s to 100000 m/s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 m/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- In time mode the final velocity rises one-for-one with it and the distance rises by u × t.
- Get it right
- Convert km/h to m/s by dividing by 3.6. Only zero or forward (downward) speeds are accepted, so a body thrown upwards is not covered.
What this means: Acceleration (negative to decelerate)
- What it is
- The constant rate at which speed changes along the line of motion, used only in time mode. A positive value speeds the body up; a negative value slows it down, as in braking.
- Units and format
- Enter a number in m/s²: metres per second squared. The form accepts -100000 m/s² to 100000 m/s². Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 m/s² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Final velocity changes by a × t and distance by half of a × t².
- Get it right
- If deceleration would carry on past standstill, the tool reports the body reversing and gives the time at which it stopped.
What this means: Time
- What it is
- How long the constant acceleration lasts, from the starting speed to the moment at which you want the speed and distance; used only in time mode.
- Units and format
- Enter a number in s: seconds. The form accepts 0.000001 s to 1000000 s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use seconds: multiply a time in minutes by 60.
What this means: Final velocity (distance mode)
- What it is
- The speed the body must reach, or slow down to, at the end of the given distance; used only in distance mode to find the acceleration and time required.
- Units and format
- Enter a number in m/s: metres per second. The form accepts 0 m/s to 100000 m/s. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 m/s as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The acceleration needed rises with the difference between the squares of final and initial speed.
- Get it right
- Initial and final speed cannot both be zero. To bring the body to rest, set this to zero and give a starting speed.
What this means: Distance (distance mode)
- What it is
- The length of travel over which the speed changes from initial to final, such as a stopping distance or an acceleration ramp; used only in distance mode.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 10000000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Doubling the distance halves the acceleration needed and doubles the time for the same speeds.
What this means: Fall height (fall mode)
- What it is
- The vertical drop from the point of release to the point of impact, used only in fall mode. The body falls under standard gravity, 9.80665 m/s², with no air drag.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 100000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- From rest, impact speed rises with the square root of the height, so four times the height doubles it.
- Get it right
- Measure to where the body actually strikes, not to the floor if it lands on something higher.
Method and formulas
- v = u + a t and s = u t + ½ a t².
- a = (v² − u²) ÷ 2 s and t = 2 s ÷ (u + v).
- Falling body: v = √(u² + 2 g h), t = (v − u) ÷ g, with g = 9.80665 m/s².
- Example: from rest at 2 m/s² for 5 s a body reaches 10 m/s after 25 m.
Limitations
- Constant acceleration only; real drives ramp acceleration (jerk limits).
- Air resistance is neglected for falling bodies.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a conveyor or slide acceleration ramp
- Check stopping distance for a decelerating carriage
- Estimate the impact speed of a dropped part
Who it is for
- Design engineers
- Mechanical engineers
- Engineering students
What you get
- Final velocity
- Distance travelled or acceleration needed
- Time taken
Interpreting the result
- For a fall the impact energy is m g h; use it with an impact or shock-load tool to size the stop.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 175–181. Topic reference; no source tables or recommended settings are reproduced.
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Open the Uniformly Accelerated Motion and Falling Bodies calculator