#math/calculus A parametric curve becomes a physical motion when $t$ represents time. The first derivative records both direction and rate of travel; its magnitude keeps only the rate, while the second derivative records how the velocity changes. If $\mathbf r(t)$ is position, then $ \mathbf v(t)=\mathbf r'(t),\qquad \text{speed}=\|\mathbf v(t)\|,\qquad \mathbf a(t)=\mathbf r''(t). $ When $\mathbf v(t_0)\ne0$, the tangent line is $ \ell(s)=\mathbf r(t_0)+s\mathbf v(t_0). $ For the helix $\mathbf r(t)=\langle\cos t,\sin t,t\rangle$, $ \mathbf v=\langle-\sin t,\cos t,1\rangle,\qquad \mathbf a=\langle-\cos t,-\sin t,0\rangle, $ and the speed is constant: $\|\mathbf v\|=\sqrt2$. Acceleration points horizontally toward the helix's axis even though the particle continues upward. Given acceleration and initial data, integrate componentwise. If $\mathbf a=-\mathbf k$, $\mathbf r(0)=\langle0,0,1\rangle$, and $\mathbf v(0)=\langle1,1,0\rangle$, then $ \mathbf r(t)=\left\langle t,t,1-\frac{t^2}{2}\right\rangle. $ Speed accumulated over time gives [[14G1 - Arc Length and Unit-Speed Parametrization|arc length]].