Formula Sheet · Curvilinear Motion

Vectors and the path

Definitions

\[ \begin{aligned} \vvec &= \frac{d\rvec}{dt} \\ \avec &= \frac{d\vvec}{dt} = \frac{d^2\rvec}{dt^2} \end{aligned} \]

Speed

\[ v = |\vvec| = \frac{ds}{dt} \]

\(\vvec\) is tangent to the path, in the direction of motion.
\(\avec\) is tangent to the hodograph, not the path; on a curve it points to the concave side.
Displacement \(\Delta\rvec\) is a chord; distance \(\Delta s \ge |\Delta\rvec|\). Average velocity \(\Delta\rvec/\Delta t\) \(\ne\) average speed \(\Delta s/\Delta t\).
Rectilinear review: \(v = \dot s\), \(a = \dot v\), \(a\,ds = v\,dv\); constant \(a_c\): \(v = v_0 + a_c t\), \(s = s_0 + v_0 t + \tfrac12 a_c t^2\), \(v^2 = v_0^2 + 2a_c(s - s_0)\).

Rectangular components

\(\ihat\), \(\jhat\) fixed: differentiate the coordinates

\[ \begin{aligned} \rvec &= x\,\colX{\ihat} + y\,\colY{\jhat} \\ \vvec &= \dot x\,\colX{\ihat} + \dot y\,\colY{\jhat} \\ \avec &= \ddot x\,\colX{\ihat} + \ddot y\,\colY{\jhat} \end{aligned} \]

\(v = \sqrt{v_x^2 + v_y^2}\), direction \(\theta_v = \atantwo(v_y, v_x)\) (the slope angle of the path).
Path \(y = f(x)\): \(v_y = f'(x)\,v_x\), \(\ a_y = f''(x)\,v_x^2 + f'(x)\,a_x\).
From \(\avec(t)\): \(v_x = v_{x0} + \int a_x\,dt\), \(x = x_0 + \int v_x\,dt\) (same for \(y\)).

Projectile motion

\[ \begin{aligned} a_x &= 0, & a_y &= -g \\ v_x &= v_0\cos\theta_0, & v_y &= v_0\sin\theta_0 - g t \\ x &= x_0 + v_{x0}t, & y &= y_0 + v_{y0}t - \tfrac12 g t^2 \end{aligned} \] \[ v_y^2 = v_{y0}^2 - 2g(y - y_0) \]

Level ground only (lands at launch height)

\[ T = \frac{2v_0\sin\theta_0}{g}, \quad h_\text{max} = \frac{v_0^2\sin^2\theta_0}{2g}, \quad R = \frac{v_0^2\sin 2\theta_0}{g} \]

Trajectory (origin at launch): \(y = x\tan\theta_0 - \dfrac{g x^2}{2v_0^2\cos^2\theta_0}\), with \(\tfrac{1}{\cos^2\theta_0} = 1 + \tan^2\theta_0\).
At the top: \(v_y = 0\), \(\vvec\) horizontal, \(a = g\) down, \(\rho = v_{x0}^2/g\).
Strategy: get \(t\) from one direction, use it in the other.

Path (\(n\)–\(t\)) coordinates

Figure F.1 \(\et\) along the motion; \(\en\) toward the center of curvature \(C\).

Velocity and acceleration

\[ \vvec = v\,\colT{\et}, \qquad \avec = a_t\,\colT{\et} + a_n\,\colN{\en} \] \[ a_t = \dot v = v\frac{dv}{ds}, \qquad a_n = \frac{v^2}{\rho}, \qquad |\avec| = \sqrt{a_t^2 + a_n^2} \]

\(v_n = 0\) always. \(a_n \ge 0\), always toward the center of curvature.
\(d\et/dt = (v/\rho)\,\en\): this is where \(a_n\) comes from.
Straight line: \(\rho = \infty\), \(a_n = 0\). Constant speed: \(a_t = 0\).
Circle, radius \(r\): \(v = \omega r\), \(a_t = \alpha r\), \(a_n = \omega^2 r = v\omega\).

Radius of curvature

Path \(y = f(x)\)

\[ \rho = \frac{\left[1 + (dy/dx)^2\right]^{3/2}}{|d^2y/dx^2|} \]

Motion \(x(t), y(t)\)

\[ \rho = \frac{v^2}{a_n} = \frac{(\dot x^2 + \dot y^2)^{3/2}}{|\dot x\ddot y - \dot y\ddot x|} \]

Crest or sag (\(y' = 0\)): \(\rho = 1/|y''|\). Parabola \(y = kx^2\) at its vertex: \(\rho = 1/(2|k|)\).
Inflection point (\(y'' = 0\)): \(\rho = \infty\); \(\en\) switches sides.
Center of curvature above the path if \(y'' \gt 0\), below if \(y'' \lt 0\).

Converting between systems

\(x\)–\(y\) → \(n\)–\(t\)

\[ \begin{aligned} \et &= \vvec/v \\ a_t &= \frac{v_x a_x + v_y a_y}{v} \\ a_n &= \frac{|v_x a_y - v_y a_x|}{v} \end{aligned} \]

\(n\)–\(t\) → \(x\)–\(y\) (turning left)

\[ \begin{aligned} a_x &= a_t\cos\psi - a_n\sin\psi \\ a_y &= a_t\sin\psi + a_n\cos\psi \end{aligned} \]

\(\psi\) = direction of travel from \(+x\). Turning right: change the sign of the \(a_n\) terms.

\(\avec\cdot\vvec \gt 0\): speeding up; \(\lt 0\): slowing down; \(= 0\): steady speed.
\(v_x a_y - v_y a_x \gt 0\): turning counter-clockwise (left); \(\lt 0\): clockwise (right).
Check: \(a_x^2 + a_y^2 = a_t^2 + a_n^2\).

Choosing a system and design limits

Given	Use
\(x(t)\), \(y(t)\); gravity only; \(a_x\), \(a_y\) separately	\(x\)–\(y\)
Path \(y = f(x)\) with \(v_x\) known	\(x\)–\(y\), chain rule
Path and speed along it; circular motion	\(n\)–\(t\)
"How sharply does it turn?" "Speeding up?"	\(n\)–\(t\) (convert)

Minimum curve radius: \(\rho_\text{min} = v^2/a_{n,\max}\).
Crest (center below): contact lost when \(v^2/\rho \gt g\), so \(v_\text{max} = \sqrt{g\rho}\).
\(g\)-load: \(a_n/g\). Convert km/h to m/s by dividing by 3.6.

Common mistakes

"Constant speed means no acceleration." On a curve \(a_n = v^2/\rho \ne 0\).
Adding perpendicular parts as numbers. \(|\avec| = \sqrt{a_t^2 + a_n^2}\), not \(a_t + a_n\).
A normal velocity. \(\vvec = v\,\et\); there is no \(v_n\).
Dropping \(f''\). On \(y = f(x)\), \(a_y = f''v_x^2 + f'a_x\), even when \(a_x = 0\).
The range formula off a cliff. \(R = v_0^2\sin 2\theta_0/g\) needs landing height = launch height.
\(v = 0\) at the top of a projectile. Only \(v_y = 0\); \(a = g\) all the way.
Constant-acceleration formulas for variable \(a_t\). Use \(dv = a_t\,dt\) or \(v\,dv = a_t\,ds\).
Mixing component sets. Never combine \(a_x\) with \(a_n\) or \(a_t\).
Quadrant of \(\theta_v\). \(\arctan(v_y/v_x)\) is off by \(180^\circ\) when \(v_x \lt 0\).