Library / Foundations: Physics / FPH-03
Foundations: Physics
Final speed from distance (v² = u² + 2as)
Speed after a given distance at constant acceleration.
- Ref
- FPH-03
- Rev
- v1
- Field
- Foundations: Physics
- Result
- v [m/s]
For straight-line motion with constant acceleration a. u and v are initial and final velocities and s is displacement.
This links start speed, end speed, acceleration and distance without needing the time.
Calculator
m/s
m/s
m/s²
m
- Speed in km/h74.22 km/h
- Time taken (v − u)/a7.808 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
Final speed (m/s) current value 20.62 m/s
sDistanceKey driver
5.3963.4
aAcceleration
6.7145
uInitial speed
2036.1
- Stopping distance grows with the square of speed: doubling speed quadruples braking distance.
- For braking, use a negative acceleration and solve for the distance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Distance | 1 to 1,000 m | 5.385 | 63.44 | 58.06 | +10% | +4.6 % |
| a Acceleration | 0.1 to 10 m/s² | 6.708 | 45 | 38.29 | +10% | +4.6 % |
| u Initial speed | 0 to 30 m/s | 20 | 36.06 | 16.06 | +10% | +0.616 % |
Sweep of Distance
| Distance (m) | Final speed (m/s) |
|---|---|
| 1 | 5.385 |
| 2.371 | 5.872 |
| 5.623 | 6.892 |
| 13.34 | 8.851 |
| 31.62 | 12.31 |
| 74.99 | 18.03 |
| 177.8 | 27.14 |
| 421.7 | 41.37 |
| 1,000 | 63.44 |
Sweep one variable
Final speed (m/s)Hover, touch, or use the arrow keys on the chart
Variables and how each one contributes
v
Final speed
What the equation gives you. Everything else feeds it.
u
Initial speed
Speed at the start. Enters squared.
a
Acceleration
Constant acceleration. Raises v with the square root of a.
s
Distance
Distance covered. Raises v with the square root of s.
Worked example with the values above
- u² = 25; 2as = 2 × 2 × 100 = 400.
- v² = 425.
- v = 20.62 m/s.
Assumptions and limits
- Constant acceleration.
- Straight-line motion.
Common mistakes
- Forgetting to take the square root.
- Wrong sign for deceleration.
Statistics and uncertainty
The square root halves relative errors in a and s: δv/v ≈ ½√[(δa/a)² + (δs/s)²] when u = 0.
Where it is used
- Stopping distances.
- Speed at the end of a launch rail or runway.
Maths you need
- Squares and roots: v² and √
- Acceleration: constant rate of change of velocity