Motion profile (s-v-a) calculator
Calculate uniformly accelerated motion: enter three of the five quantities distance, initial and final velocity, acceleration and time, and the calculator fills in the rest – translational or rotational. In trapezoidal mode it derives the full s, v and a curves plus total distance and total time from peak velocity, acceleration, constant-travel time and deceleration, live with every input.
Motion profile calculator
Model: uniformly accelerated motion with constant acceleration (no jerk limitation). For jerk-limited S-curve profiles compute the phases individually. For full motor sizing with load, gearbox and RMS torque use the drive sizing calculator.
Results
Calculating …
Formulas and fundamentals
The basis is the three equations of uniformly accelerated motion with constant acceleration a: velocity grows linearly as v_e = v_a + a·t, distance follows as the area under velocity s = (v_a + v_e)/2·t = v_a·t + ½·a·t², and the time-free relation v_e² = v_a² + 2·a·s links the velocities directly to distance. Between the five quantities v_a, v_e, a, s and t there are thus two independent equations, so from three known quantities the remaining two follow uniquely. The calculator automatically selects the matching formula and takes roots non-negative (physical velocity magnitudes).
Rotational motion is fully analogous: distance, velocity and acceleration are replaced by angle of rotation φ, angular velocity ω and angular acceleration α, with ω_e = ω_a + α·t, φ = (ω_a + ω_e)/2·t and ω_e² = ω_a² + 2·α·φ. Conversion uses the effective radius or the feed constant K_VA: v = ω·K_VA/(2π), or for a wheel v = ω·r. Angles are handled internally in radians; one revolution is 2π rad, a rotational speed n in rpm corresponds to the angular velocity ω = 2π·n/60.
The trapezoidal profile is the standard positioning case: an acceleration phase with a from v_a = 0 up to the peak velocity v_max, a constant-travel phase at v_max and a deceleration phase with −a (or a separate deceleration) back to zero. The phase times are t_B = v_max/a and t_V = v_max/a_V, the phase distances s_B = v_max²/(2·a), s_K = v_max·t_K and s_V = v_max²/(2·a_V). Total distance and total time are the sums of the phases. The velocity curve forms the eponymous trapezoid, the acceleration curve a rectangle with signs +a, 0 and −a_V, and the distance curve a parabolic-linear shape.
Worked example
A slide is to accelerate from 3 m/s to 5 m/s at a uniform 2 m/s². From v_e = v_a + a·t the acceleration time is t = (v_e − v_a)/a = (5 − 3)/2 = 1.0 s. The distance travelled is s = (v_a + v_e)/2·t = (3 + 5)/2·1.0 = 4.0 m; equivalently via the time-free relation s = (v_e² − v_a²)/(2·a) = (25 − 9)/4 = 4.0 m.
If instead the initial velocity v_a = 3 m/s, the time t = 5 s and the acceleration a = 1.5 m/s² are known, the distance is s = v_a·t + ½·a·t² = 15 + 18.75 = 33.75 m and the final velocity v_e = v_a + a·t = 3 + 7.5 = 10.5 m/s.
A trapezoidal profile with v_max = 1 m/s, acceleration a = 2 m/s² and constant-travel time t_K = 1 s has phase times t_B = t_V = 0.5 s, hence a total time of 2 s. The phase distances are s_B = s_V = 0.25 m and s_K = 1 m, for a total distance of 1.5 m.
Frequently asked questions
How many quantities do I need to enter?
In solver mode exactly three of the five quantities initial velocity, final velocity, acceleration, distance and time are required. Because two independent equations relate them, the remaining two are then uniquely determined. With only two entries the motion is underdetermined; with four it may be overdetermined (the values must then be physically consistent).
Does the calculation apply only to constant acceleration?
Yes. The formulas assume an acceleration a (or angular acceleration α) that is constant over the phase considered. For jerk-limited profiles (S-curves with finite jerk) or varying acceleration they hold only piecewise; each phase with constant a can, however, be computed this way and chained together – which is exactly what the trapezoidal mode does.
What is the difference between solver and trapezoidal mode?
The solver treats a single phase of constant acceleration and fills in missing quantities. The trapezoidal mode sets the initial velocity to zero and combines three phases – accelerate, cruise, decelerate – into a complete positioning move with total distance, total time and the s-v-a curves versus time.
How do I convert between rotational speed and angular velocity?
Via ω = 2π·n/60 with n in rpm and ω in rad/s. One full revolution equals 2π ≈ 6.283 rad. To convert to linear motion use the feed constant K_VA (distance per revolution): v = ω·K_VA/(2π); for a wheel of diameter d, K_VA = π·d and thus v = ω·r with r = d/2.
Why can the result be under- or overdetermined?
Underdetermined means fewer than three quantities are given – then there are infinitely many solutions. Overdetermined means more than three are given, which may contradict each other; the calculator then computes from the first consistent quantities and ignores contradictions. For a unique result, enter exactly three mutually consistent quantities.
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