Library / Mechanical / MEC-01
Mechanical
Constant-acceleration displacement
Distance travelled in a given time at constant acceleration.
- Ref
- MEC-01
- Rev
- v1
- Field
- Mechanical
- Result
- s [m]
s in metres, u the initial velocity in m/s, a in m/s², t in seconds. One-dimensional motion along a line, s signed displacement. The companion relation is v = u + at.
Starting at speed u and speeding up at a steady rate a, the distance covered grows with time faster and faster: a part from the starting speed, plus a part from the acceleration that grows with the square of time.
Calculator
- Final velocity v = u + at26 m/s
- Average velocity18 m/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.
- Distance is quadratic in time. A vehicle that takes 10 s to cover a distance needs only about 14 s (not 20 s) for twice the distance from rest.
- Starting from rest (u = 0) the formula reduces to s = ½at².
- The average velocity is the mean of the start and end velocities, which is why distance is average speed × time.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| t Elapsed time | 1 to 30 s | 11 | 1,200 | 1,189 | +10% | +14.9 % |
| a Acceleration | 0.1 to 10 m/s² | 83.2 | 400 | 316.8 | +10% | +4.44 % |
| u Initial velocity | 0 to 30 m/s | 64 | 304 | 240 | +10% | +5.56 % |
Sweep of Elapsed time
| Elapsed time (s) | Displacement (m) |
|---|---|
| 1 | 11 |
| 1.53 | 17.64 |
| 2.34 | 28.88 |
| 3.58 | 48.62 |
| 5.477 | 84.77 |
| 8.379 | 154 |
| 12.82 | 292.5 |
| 19.61 | 580.7 |
| 30 | 1,200 |
Sweep one variable
Variables and how each one contributes
Displacement
What the equation gives you. Everything else feeds it.
Initial velocity
Velocity at the start of the interval. Adds u·t: proportional to time, and the larger the time the larger its share.
Acceleration
Constant acceleration along the line. Adds ½at²: proportional to a and dominant for long times.
Elapsed time
Time since the start of the interval. The strongest driver: distance grows with t², so doubling the time more than doubles the distance.
Worked example with the values above
- Speed part: ut = 10 × 8 = 80 m.
- Acceleration part: ½at² = 0.5 × 2 × 8² = 64 m.
- Add them: s = 144 m.
Assumptions and limits
- Acceleration is constant for the whole interval.
- Motion along a straight line.
- Signs follow one chosen positive direction.
Common mistakes
- Using the formula when acceleration changes (e.g. engine thrust falling off).
- Dropping the sign for deceleration.
- Mixing km/h with m/s.
Statistics and uncertainty
Time enters squared, so a relative timing error is doubled in the acceleration term: for a from rest, δs/s ≈ √[(δa/a)² + (2δt/t)²]. A 1% error in timing gives about 2% in distance.
Where it is used
- Take-off and landing runs.
- Braking distance estimates.
- Launch-rail length for a given exit speed.
Maths you need
- Velocity and acceleration: rates of change of position and velocity
- Squares: time enters as t²