Reference · 42 terms

Glossary

Short, precise definitions of the words and symbols used in this module, each linked to the lesson where it is taught. Filter the list as you type, jump to a letter, or look a symbol up in Symbols at a glance.

Showing all 42 terms

A

Acceleration \(\avec\): The rate of change of the velocity vector, \(\avec = d\vvec/dt\), in m/s². It is nonzero whenever the speed or the direction of motion changes, and on a curve it points toward the concave side of the path. In rectangular components \(\avec = \ddot x\,\ihat + \ddot y\,\jhat\); in path components \(\avec = \dot v\,\et + (v^2/\rho)\,\en\).; See: Lesson 2Related: Tangential acceleration \(a_t\), Normal acceleration \(a_n\), Hodograph
Angular acceleration \(\alpha\): The rate of change of angular velocity, \(\alpha = \dot\omega\), in rad/s². For a point at distance \(r\) from an axis it produces the tangential acceleration \(a_t = \alpha r\).; See: Lesson 6Related: Angular velocity \(\omega\), Tangential acceleration \(a_t\)
Angular velocity \(\omega\): The rate at which a body turns, in rad/s (\(N\) rpm = \(2\pi N/60\) rad/s). A point at distance \(r\) from the axis moves at \(v = \omega r\) and has normal acceleration \(a_n = \omega^2 r\).; See: Lesson 6Related: Angular acceleration \(\alpha\), Centripetal acceleration
atan2: A two-argument arctangent, \(\atantwo(v_y, v_x)\), that returns the direction of a vector in the correct quadrant. \(\arctan(v_y/v_x)\) alone is off by \(180^\circ\) when \(v_x \lt 0\).; See: Lesson 3Related: Direction of motion \(\psi\)
Average speed: Distance travelled along the path divided by the time taken, \(\Delta s/\Delta t\). A scalar; it is never less than the magnitude of the average velocity.; See: Lesson 2Related: Average velocity, Distance travelled \(\Delta s\)
Average velocity: Displacement divided by time, \(\vvec_\text{avg} = \Delta\rvec/\Delta t\). It points along the chord between the start and end positions and is zero after a full lap.; See: Lesson 2Related: Average speed, Displacement \(\Delta\rvec\), Velocity \(\vvec\)

C

Center of curvature: The center \(C\) of the osculating circle at a point of the path. It lies on the concave side, at distance \(\rho\) from the path, in the direction of \(\en\).; See: Lesson 5Related: Osculating circle, Radius of curvature \(\rho\), Normal unit vector \(\en\)
Centripetal acceleration: Another name for the normal acceleration \(a_n = v^2/\rho\): the part of the acceleration that points toward the center of curvature and turns the velocity. For circular motion it is \(\omega^2 r\).; See: Lesson 6Related: Normal acceleration \(a_n\)
Concave side: The inner side of a bend: the side the path curves toward. The center of curvature, \(\en\) and the normal part of the acceleration are all on the concave side.; See: Lesson 2Related: Center of curvature, Normal unit vector \(\en\)
Crest (vertical curve): The top of a hill in a road or track, where the center of curvature is below. A vehicle keeps contact only while \(v^2/\rho \le g\), so \(v_\text{max} = \sqrt{g\rho}\).; See: Lesson 9Related: Sag, \(g\)-load
Curvature \(\kappa\): How sharply a path bends: \(\kappa = 1/\rho = d\psi/ds\), the rate at which the direction of the tangent turns per unit length of path. A straight line has \(\kappa = 0\).; See: Lesson 5Related: Radius of curvature \(\rho\)
Curvilinear motion: Motion of a particle along a curved path. When the whole path lies in one plane it is plane curvilinear motion, the subject of this module.; See: Lesson 1Related: Rectilinear motion, Path

D

Direction of motion \(\psi\): The angle of the velocity (and of \(\et\)) measured counter-clockwise from the \(+x\) axis: \(\et = \cos\psi\,\ihat + \sin\psi\,\jhat\). It equals the slope angle of the path.; See: Lesson 5Related: Tangential unit vector \(\et\), atan2
Displacement \(\Delta\rvec\): The change in position between two instants, \(\Delta\rvec = \rvec(t + \Delta t) - \rvec(t)\): the straight arrow (chord) from the first position to the second.; See: Lesson 2Related: Distance travelled \(\Delta s\), Position vector \(\rvec\)
Distance travelled \(\Delta s\): The length of path covered, a positive scalar. On a curve it is longer than the displacement: \(\Delta s \ge |\Delta\rvec|\).; See: Lesson 2Related: Displacement \(\Delta\rvec\), Path coordinate \(s\)
Dot notation: A dot over a quantity means its time derivative: \(\dot x = dx/dt\), \(\ddot x = d^2x/dt^2\), \(\dot v = dv/dt\).; See: Lesson 1Related: Velocity \(\vvec\), Acceleration \(\avec\)

E

\(\hat{\mathbf{e}}_t,\ \hat{\mathbf{e}}_n\): Meriam and Kraige's notation for the path unit vectors, which this module writes \(\et, \en\) (Hibbeler's notation). Same directions, different letters.; See: Lesson 1Related: Tangential unit vector \(\et\), Normal unit vector \(\en\)

G

\(g\)-load: An acceleration expressed as a multiple of \(g\), for example \(a_n/g\) for a pilot pulling out of a dive. Used to compare motions with human and structural limits.; See: Lesson 9Related: Gravitational acceleration \(g\), Normal acceleration \(a_n\)
Gravitational acceleration \(g\): The acceleration of a freely falling body near the Earth's surface, \(g = 9.81\ \text{m/s}^2\), straight down. With \(y\) up, a projectile has \(a_y = -g\).; See: Lesson 4Related: Projectile, \(g\)-load

H

Hodograph: The curve traced by the tip of the velocity vector when every velocity is drawn from one fixed point. The acceleration is tangent to the hodograph, just as the velocity is tangent to the path.; See: Lesson 2Related: Acceleration \(\avec\), Velocity \(\vvec\)

I

Inflection point: A point where a path changes from bending one way to bending the other (\(y'' = 0\)). There \(\rho = \infty\), \(a_n = 0\), and \(\en\) jumps to the other side of the path.; See: Lesson 5Related: Radius of curvature \(\rho\), Normal unit vector \(\en\)

M

Maximum height (of a projectile): The highest point of a trajectory, where \(v_y = 0\). Above the launch point it is \(h_\text{max} = v_0^2\sin^2\theta_0/(2g)\).; See: Lesson 4Related: Projectile, Range

N

Normal acceleration \(a_n\): The component of the acceleration along \(\en\), \(a_n = v^2/\rho\). It is never negative, points toward the center of curvature, and changes the direction of the velocity.; See: Lesson 6Related: Tangential acceleration \(a_t\), Centripetal acceleration, Radius of curvature \(\rho\)
Normal unit vector \(\en\): The unit vector perpendicular to the path, pointing toward the center of curvature (the concave side). Reversing the direction of travel does not change it. Meriam and Kraige write it \(\hat{\mathbf{e}}_n\).; See: Lesson 5Related: Tangential unit vector \(\et\), Center of curvature, \(\hat{\mathbf{e}}_t,\ \hat{\mathbf{e}}_n\)

O

Osculating circle: The circle that fits a curve best at a point: it has the same tangent and bends at the same rate. Its radius is the radius of curvature and its center is the center of curvature.; See: Lesson 5Related: Radius of curvature \(\rho\), Center of curvature

P

Path: The curve traced by the tip of the position vector as the particle moves. The velocity is always tangent to it.; See: Lesson 2Related: Position vector \(\rvec\), Trajectory equation
Path coordinate \(s\): Distance measured along the path from a fixed point, increasing in the direction of motion. Its rate of change is the speed, \(v = ds/dt\).; See: Lesson 5Related: Speed \(v\), Distance travelled \(\Delta s\)
Path coordinates (normal and tangential, \(n\)–\(t\)): Components along \(\et\) and \(\en\), unit vectors that ride with the particle. They give \(\vvec = v\,\et\) and \(\avec = \dot v\,\et + (v^2/\rho)\,\en\). Best when the path and the speed along it are known.; See: Lesson 5Related: Rectangular components, Tangential unit vector \(\et\), Normal unit vector \(\en\)
Position vector \(\rvec\): The arrow from a fixed origin to the particle. In rectangular components \(\rvec = x\,\ihat + y\,\jhat\).; See: Lesson 2Related: Displacement \(\Delta\rvec\), Path
Projectile: A particle moving under gravity alone. With no air resistance, \(a_x = 0\) and \(a_y = -g\): constant horizontal velocity and constant downward acceleration, sharing only the time.; See: Lesson 4Related: Range, Time of flight, Trajectory equation

R

Radius of curvature \(\rho\): The radius of the osculating circle. For \(y = f(x)\): \(\rho = (1 + y'^2)^{3/2}/|y''|\); for a motion: \(\rho = v^2/a_n = v^3/|\dot x\ddot y - \dot y\ddot x|\). A straight line has \(\rho = \infty\).; See: Lesson 7Related: Curvature \(\kappa\), Osculating circle, Normal acceleration \(a_n\)
Range (of a projectile): The horizontal distance to the landing point. On level ground (landing at launch height) \(R = v_0^2\sin 2\theta_0/g\), largest at \(45^\circ\).; See: Lesson 4Related: Projectile, Time of flight
Rectangular components (\(x\)–\(y\)): Components along the fixed directions \(\ihat\) and \(\jhat\): \(\vvec = \dot x\,\ihat + \dot y\,\jhat\), \(\avec = \ddot x\,\ihat + \ddot y\,\jhat\). Best when \(x(t)\) and \(y(t)\), or \(a_x\) and \(a_y\), are known separately.; See: Lesson 3Related: Path coordinates
Rectilinear motion: Motion along a straight line, described by one coordinate \(s\): \(v = \dot s\), \(a = \dot v\) and \(a\,ds = v\,dv\).; See: Lesson 1Related: Curvilinear motion

S

Sag (vertical curve): The bottom of a dip in a road or track, where the center of curvature is above and occupants feel heavier. Contact is never lost in a sag.; See: Lesson 7Related: Crest
Speed \(v\): The magnitude of the velocity, \(v = |\vvec| = ds/dt\), in m/s. A scalar, never negative.; See: Lesson 2Related: Velocity \(\vvec\), Path coordinate \(s\)

T

Tangential acceleration \(a_t\): The component of the acceleration along \(\et\): \(a_t = \dot v = v\,dv/ds\). Positive when the particle speeds up, negative when it slows down, zero at constant speed.; See: Lesson 6Related: Normal acceleration \(a_n\), Angular acceleration \(\alpha\)
Tangential unit vector \(\et\): The unit vector tangent to the path in the direction of motion, \(\et = \vvec/v\). It turns as the particle moves: \(d\et/dt = (v/\rho)\,\en\). Meriam and Kraige write it \(\hat{\mathbf{e}}_t\).; See: Lesson 5Related: Normal unit vector \(\en\), \(\hat{\mathbf{e}}_t,\ \hat{\mathbf{e}}_n\)
Time of flight: How long a projectile is in the air. On level ground \(T = 2v_0\sin\theta_0/g\); otherwise solve \(y(t) = y_\text{landing}\) for the positive root.; See: Lesson 4Related: Projectile, Range
Trajectory equation: The path of a projectile with \(t\) eliminated: \(y = x\tan\theta_0 - g x^2/(2v_0^2\cos^2\theta_0)\), a downward parabola. Solving it for \(\tan\theta_0\) aims at a target.; See: Lesson 4Related: Projectile, Path

U

Unit vector: A vector of length 1 that marks a direction. This module writes unit vectors in bold, as Hibbeler does; some books add a hat. \(\ihat, \jhat\) are fixed; \(\et, \en\) turn with the particle.; See: Lesson 1Related: Tangential unit vector \(\et\), Normal unit vector \(\en\)

V

Velocity \(\vvec\): The rate of change of position, \(\vvec = d\rvec/dt\), in m/s. Always tangent to the path in the direction of motion, with magnitude equal to the speed. In path coordinates \(\vvec = v\,\et\).; See: Lesson 2Related: Speed \(v\), Acceleration \(\avec\), Average velocity

Symbols at a glance

Met a symbol in a lesson or in another textbook and not sure what it stands for? Find it here, then follow the link to its entry.

Symbols used in this module and in other books
Symbol	Meaning	Entry
\(\rvec\)	Position vector	Position vector \(\rvec\)
\(\vvec,\ v\)	Velocity and speed	Velocity \(\vvec\)
\(\avec\)	Acceleration	Acceleration \(\avec\)
\(\ihat,\ \jhat\)	Fixed \(x\) and \(y\) unit vectors	Unit vector
\(\colT{\et},\ \colN{\en}\)	Tangential and normal unit vectors	Path coordinates
\(\hat{\mathbf{e}}_t,\ \hat{\mathbf{e}}_n\)	Meriam and Kraige's \(\et, \en\)	\(\hat{\mathbf{e}}_t,\ \hat{\mathbf{e}}_n\)
\(v_x, v_y,\ a_x, a_y\)	Rectangular components	Rectangular components
\(a_t,\ a_n\)	Tangential and normal accelerations	Tangential acceleration \(a_t\)
\(\rho,\ \kappa\)	Radius of curvature, curvature \(1/\rho\)	Radius of curvature \(\rho\)
\(s\)	Distance along the path	Path coordinate \(s\)
\(\psi,\ \theta_v\)	Direction of motion from \(+x\)	Direction of motion \(\psi\)
\(\theta_0,\ v_0\)	Launch angle and launch speed	Projectile
\(T,\ R,\ h_\text{max}\)	Time of flight, range, maximum height	Range
\(\omega,\ \alpha\)	Angular velocity and acceleration	Angular velocity \(\omega\)
\(g\)	\(9.81\ \text{m/s}^2\), downward	Gravitational acceleration \(g\)
\(\dot x,\ \ddot x\)	First and second time derivatives	Dot notation