Velocity, speed, and acceleration

If a particle’s position at time $t$ is given by the vector-valued function $\mathbf{r}(t)$, then

• Velocity is $\mathbf{v}(t)={\mathbf{r}}^{\prime }(t)$ — a vector, direction of motion times speed.
• Speed is $|\mathbf{v}(t)|=|{\mathbf{r}}^{\prime }(t)|$ — a scalar, how fast.
• Acceleration is $\mathbf{a}(t)={\mathbf{v}}^{\prime }(t)={\mathbf{r}}^{\prime \prime }(t)$ — a vector, rate of change of velocity.

This is the kinematic structure of motion in 3D.

Difference between speed and velocity

In everyday speech the two words are often interchanged, but in vector calculus they’re sharply different: velocity is a vector (with a direction), speed is its magnitude (a non-negative scalar). A particle moving in a circle at constant speed has velocity that keeps changing direction — so it’s accelerating, even though speed is constant.

Worked example: the helix

$\mathbf{r}(t)=⟨\cos t,\sin t,t⟩$.

$\mathbf{v}(t)=⟨-\sin t,\cos t,1⟩$.

Speed: $|\mathbf{v}|=\sqrt{{\sin }^{2}t+{\cos }^{2}t+1}=\sqrt{2}$. Constant.

$\mathbf{a}(t)=⟨-\cos t,-\sin t,0⟩$. Magnitude $|\mathbf{a}|=1$.

The acceleration points horizontally inward, toward the $z$-axis — centripetal acceleration of circular motion in the $xy$-plane, happening simultaneously with the steady $\hat{\mathbf{z}}$ drift. Constant speed, nonzero acceleration: the helix is a clean illustration of why those two concepts must be distinguished.

Displacement vs. distance traveled

Two different integrals.

Net displacement from time $a$ to $b$:

${\int }_a^b\mathbf{v}(t)dt=\mathbf{r}(b)-\mathbf{r}(a),$

a vector. This is the straight-line vector from start to end, ignoring the path.

Distance traveled (path length) from $a$ to $b$:

${\int }_a^b|\mathbf{v}(t)|dt,$

a non-negative scalar. This is the Arc length of the path.

For a particle that returns to its starting point, the net displacement is $0$ but the distance traveled is the total path length — usually nonzero.

Decomposition of acceleration

Acceleration splits into tangential and normal components:

$\mathbf{a}=a_T\mathbf{T}+a_N\mathbf{N},$

where $\mathbf{T}$ is the unit tangent, $\mathbf{N}$ is the principal normal, $a_T=\frac{d}{dt}|\mathbf{v}|$ (rate of change of speed), and $a_N=|\mathbf{v}{|}^2/R$ with $R$ the radius of curvature (centripetal).

For uniform circular motion, $a_T=0$ (constant speed) and $a_N=v^2/R$ — pure centripetal acceleration. For straight-line motion with changing speed, $a_N=0$ and $a_T=dv/dt$ — pure tangential.