Speed describes movement right now.

Velocity includes both speed and direction. Acceleration describes how velocity changes over time. An upward-moving car can accelerate upward while speeding up or accelerate downward while slowing down.

Deceleration is a familiar name for reducing speed, rather than a separate physical quantity.

A change in speed takes time.

From rest to 1 m/s in two seconds

Average acceleration is (1 − 0) ÷ 2 = 0.5 m/s². This average does not tell us whether acceleration changed smoothly during those two seconds.

Jerk describes changing acceleration.

Jerk is the rate of change of acceleration, measured in m/s³. Abrupt changes in acceleration can feel like jolts even when the maximum speed is modest.

A shaped motion profile introduces and removes acceleration progressively. This helps manage comfort and mechanical loading.

Read velocity, acceleration and jerk together.

Read the three plots from top to bottom: how fast the car moves, how its velocity changes, and how its acceleration changes. Each uses the same two seconds along the bottom. Upwards is positive.

This example begins at rest and reaches an upward velocity of 1 m/s. It shows only the start of a journey; braking and the final stop are not plotted.

One start, three connected graphs

The same two-second start, shown as velocity, acceleration and jerk All three graphs share time zero, one and two seconds. Upward velocity rises from zero to 0.5 metres per second at one second, then to one metre per second at two seconds. Acceleration rises linearly from zero to one metre per second squared, then returns linearly to zero. Jerk is plus one metre per second cubed during the first second and minus one during the second. Open endpoints on the jerk lines show that its exact switching instants are not assigned a derivative value. Velocity (m/s) Acceleration (m/s²) Jerk (m/s³) 10.50 10.50 +10−1 0 s1 s2 s 0 s1 s2 s 0 s1 s2 s
  1. Time is in seconds on every horizontal axis. Look vertically at the same time in all three plots. The vertical axes measure different quantities, so their heights cannot be compared as if they had the same units.
  2. From 0 to 1 second: jerk is +1 m/s³. Acceleration builds from 0 to 1 m/s², making the velocity curve increasingly steep.
  3. From 1 to 2 seconds: jerk is −1 m/s³. Acceleration eases back towards zero. The car still speeds up because acceleration remains positive until the endpoint.
  4. At 2 seconds: velocity is 1 m/s and acceleration is zero. It could now continue at constant velocity. Slowing an upward-moving car would require negative acceleration.
Original mathematical example, not lift settings. The jerk lines apply between their open endpoints: jerk has no single derivative value at the ideal switching instants. Acceleration is continuous, but jerk changes in steps.

A negative jerk does not mean the car reverses. It means acceleration is becoming less positive or more negative. Direction comes from the sign of velocity.

The average acceleration here is still 0.5 m/s², as in the earlier example, but the peak is 1 m/s². Shaping the start does not remove the need to consider peak acceleration, available torque and the distance travelled.

  1. Build speedIntroduce acceleration, then ease it away.
  2. TravelHold speed where journey distance permits.
  3. Slow and stopShape deceleration and approach the landing.
Conceptual journey stages. Short runs may never reach the normal travel speed.

An S-shaped velocity profile rounds the transitions compared with an ideal straight ramp. It does not mean the car follows an S-shaped physical path.

The numbers behind the curves.

The graph is built from two equal, one-second stages. Its values follow from the definitions of acceleration and jerk, rather than from the appearance of a hand-drawn curve.

Follow the exact model and distance calculation

Let J = 1 m/s³ be the magnitude of jerk and T = 1 s be the length of each stage. The car starts with velocity and acceleration both zero. The model assumes rest before the start and constant velocity after the two stages.

First stage: jerk is +J for 0 < t < T. Acceleration is a = Jt and velocity is v = ½Jt²; these two formulas also apply at both endpoints, 0 ≤ t ≤ T. At one second, a = 1 m/s² and v = 0.5 m/s.

Second stage: let τ = t − T, the time elapsed since the midpoint. Jerk is −J for 0 < τ < T. Acceleration is a = J(T − τ) and velocity is v = ½JT² + JTτ − ½Jτ²; these two formulas also apply at both endpoints, 0 ≤ τ ≤ T. At two seconds, a = 0 and v = 1 m/s. Acceleration and velocity agree where the stages meet; jerk has no single derivative value at the switching instants.

Check the change in velocity: the area under the acceleration graph is a triangle: ½ × 2 s × 1 m/s² = 1 m/s.

Check the distance: integrating the velocity gives JT³ ÷ 6 = 1/6 m in the first stage and 5JT³ ÷ 6 = 5/6 m in the second. Together they give 1 m. This symmetric profile happens to cover the same distance as the earlier straight velocity ramp; that is not true of every profile with the same endpoints.

Why omit a numerical jerk value at the corners? The slope of acceleration changes abruptly at 0, 1 and 2 seconds. Its derivative therefore has no single value at those instants. Between them, jerk is finite and constant. An ideal straight velocity ramp instead starts and ends with a jump in acceleration, which cannot be described by finite jerk at the jump.

Comfort depends on more than one setting.

Available torque, load balance, mechanical condition, brake timing and feedback affect how closely actual motion follows the requested profile.

Gentler acceleration can require more time or distance to reach a speed. Journey planning must consider the available travel, rather than treating comfort settings independently.

A closer look

Distance travelled is the area under a velocity–time graph. For constant acceleration from rest to 1 m/s over two seconds, the triangular area is one metre. A differently shaped profile must be assessed using its own area.

Check your understanding

Can two journeys reach the same speed with different ride comfort?

Reveal the explanation

Yes. Their acceleration and changes in acceleration can differ, as can the mechanical response. Maximum speed alone does not describe the ride.

Further reading.

Optional reading to explore this lesson in more detail.

  • PMD: motion profiles and jerk

    Compare Figure 1 and the S-curve discussion to see velocity, acceleration and jerk together. The later profile equations are an optional, more mathematical extension.