M001 · Kinematics

Motion Graph Studio

Edit position, velocity or acceleration; connect signed slopes and areas to a 3D cart, reversals and the difference between distance and displacement.

Selected time $\text{Ready}$

Reference position $\text{Ready}$

Reference signed velocity $\text{Ready}$

Reference signed acceleration $\text{Ready}$

Selected driving quantity $\text{Ready}$

Signed velocity area $\text{Ready}$

Travelled distance $\text{Ready}$

Signed acceleration area $\text{Ready}$

DRAW / SCRUB / COMPARE

Follow the signs

Driver: 0 position; 1 velocity; 2 acceleration

j

1.00\,

Duration

T

6.00\,s

Integration initial position

x_0

0.00\,m

Integration initial velocity

v_0

0.00\,\frac{\mathrm m}{\mathrm s}

Driver at knot 0

q(0T/6)

0.00\,

Driver at knot 1

q(1T/6)

2.00\,

Driver at knot 2

q(2T/6)

2.00\,

Driver at knot 3

q(3T/6)

0.00\,

Driver at knot 4

q(4T/6)

-2.00\,

Driver at knot 5

q(5T/6)

-2.00\,

Driver at knot 6

q(6T/6)

0.00\,

Replay fraction

\tau

0.40\,

Choose position, velocity or acceleration as the driver. The seven equally spaced knots form a natural cubic spline. Only the driver’s second derivative vanishes at its outer endpoints; this does not mean the cart starts or ends at rest. The spline may overshoot the knot values.

Position is twice continuously differentiable in every mode. Integrals carry their constants across each knot. Initial position applies only to velocity or acceleration input; initial velocity applies only to acceleration input. Switching mode reinterprets the knot values in the displayed units.

Blue velocity area is signed displacement; copper acceleration area is signed velocity change. Distance splits the exact position polynomial at all isolated velocity zeros and adds absolute travel. Rest intervals add no distance. The independent 480-interval trapezoid comparison retains its sampling error.

This is prescribed kinematics, not a force solver. Rail extent follows the full motion; there are no invented walls. Cart dimensions and raised vector glyphs are schematic. Blue velocity and copper acceleration have separate gains fixed for a given trajectory. All quantitative plots sit below the apparatus.

Drag a blue knot or use its keyboard-accessible slider. Drag the gold clock to inspect a prefix. Export is the model trace, not an acquired measurement. No autoplay; the plots and controls remain available without WebGL.

Prescribed teaching motion; explicit interpolation and measurement boundaries.

Start with the essentials

Focus question: Can one drawn curve determine the whole motion?
One-sentence intuition: Integrals need initial values; sign changes matter.

Core mathematical model

The connected derivatives

v=\frac{\mathrm dx}{\mathrm dt},\qquad a=\frac{\mathrm dv}{\mathrm dt}

Changing the driver changes the whole trajectory, not three independent curves.

Integrals need initial values

\begin{aligned}v(t)&=v_0+\int_0^t a(s)\,\mathrm ds\\x(t)&=x_0+\int_0^t v(s)\,\mathrm ds\end{aligned}

Signed areas may cancel after a reversal.

Distance retains return travel

D(t)=\int_0^t|v(s)|\,\mathrm ds\geq|x(t)-x(0)|

Velocity roots split the polynomial into monotone travel segments.

Natural boundary of the chosen driver

q\in\{x,v,a\},\qquad q^{\prime\prime}(0)=q^{\prime\prime}(T)=0

The input spline has continuous second derivatives; natural boundary does not mean zero speed.

Common difficulties

Model and evidence

Typical misconceptionNatural spline boundaries mean zero speed.

Better mental modelChoose position, velocity or acceleration as the driver. The seven equally spaced knots form a natural cubic spline. Only the driver’s second derivative vanishes at its outer endpoints; this does not mean the cart starts or ends at rest. The spline may overshoot the knot values.

Run the experiment

01
Select the driver
Compare Draw position, Uniform motion and Constant acceleration.
What to observe: Initial conditions apply only when integrating.
02
Edit and reverse
Drag a blue knot across zero, then scrub the gold clock.
What to observe: A negative velocity returns along the rail.
03
Separate distance and displacement
Scrub Reverse and return to its end.
What to observe: Signed displacement can vanish after nonzero travel.
04
Audit the integral
Compare exact position with the separate trapezoid trace.
What to observe: Sampling error is retained instead of forced to zero.

Core mathematical model

The connected derivatives

Integrals need initial values

Distance retains return travel

Natural boundary of the chosen driver

Common difficulties

Model and evidence

Run the experiment

Select the driver

Edit and reverse

Separate distance and displacement

Audit the integral