Notation glossary

These definitions are generated from the shared notation collection. Every entry is still draft until its wording, convention, source locator, and curriculum owner receive human review.

• Acceleration
• Launch angle
• Launch speed
• Position
• Range
• Time
• Time derivative
• Velocity

$\mathbf{a}$

Acceleration

motion.acceleration

draft

Domain

motion

Units

meters per second squared

See also

Velocity

The rate at which velocity changes with time. In this course it is held constant over a scenario, so it is the same vector at every instant.

$\explain{motion.acceleration}{\mathbf{a}} = \dfrac{d\,\explain{motion.velocity}{\mathbf{v}}}{d\explain{motion.time}{t}}$

(G.1)

Acceleration is the second time derivative of Position: velocity is the rate of change of position, and acceleration is the rate of change of velocity. The straight-line chapters assume a constant acceleration for the whole scenario. This assumption makes (1.2.1) exact, not an approximation. A positive acceleration component means that the velocity component in that direction increases. A negative acceleration component means that it decreases.

Introduced by: motion.describe-position in motion.position-and-coordinates

Used directly in: Position and Coordinate Systems, Projectile Range, Velocity Under Constant Acceleration

Source records:

• openstax-university-physics-1 — ch. 3, "Average and Instantaneous Acceleration"

$\theta$

Launch angle

motion.launch-angle

draft

Domain

motion

Units

degrees above the horizontal

The angle between a projectile's initial velocity and the horizontal ground at launch.

Measured from the horizontal, not the vertical: a low, flat launch is a small angle, and a launch straight up is 90 degrees.

Introduced by: motion.compute-range in projectiles.projectile-range

Used directly in: Projectile Range, The Optimal Launch Angle

Source records:

• openstax-university-physics-1 — ch. 4, "Projectile Motion"

$v_0$

Launch speed

motion.launch-speed

draft

Domain

motion

Units

meters per second

The object's speed at the start of the scenario being analyzed — its initial velocity in a straight-line problem, or its launch speed the instant a projectile leaves the ground.

The symbol has the same meaning for motion on a straight line and for the trajectory of a projectile: the speed of the object at the instant when Time is zero. On a straight line, it is the signed initial velocity.

Introduced by: motion.predict-velocity in motion.constant-acceleration

Used directly in: Projectile Range, The Optimal Launch Angle, Velocity Under Constant Acceleration

Source records:

• openstax-university-physics-1 — ch. 3, "Instantaneous Velocity and Speed"

$\mathbf{r}$

Position

motion.position

draft

Domain

motion

Units

meters

The object's location at elapsed time t, taken as a vector from a fixed origin so a two-dimensional motion keeps its horizontal and vertical parts separate.

$\explain{motion.position}{\mathbf{r}} = (\explain{motion.position.horizontal-coordinate}{x},\ \explain{motion.position.vertical-coordinate}{y})$

(G.2)

Symbols

$x$horizontal coordinate(meters)

$y$vertical coordinate(meters)

Velocity and acceleration are defined from position: velocity is the rate of change of position with time, and acceleration is the rate of change of velocity with time. As a vector, the position keeps its horizontal and its vertical coordinate separate. So the horizontal and the vertical motion of a projectile can be analyzed independently, with the same Time.

Introduced by: motion.describe-position in motion.position-and-coordinates

Used directly in: Position and Coordinate Systems, Projectile Range

Source records:

• openstax-university-physics-1 — ch. 3, "Time, Position, and Displacement"

$R$

Range

motion.range

draft

Domain

motion

Units

meters

Total horizontal distance a projectile covers before it returns to its launch height.

$\explain{motion.range}{R} = \dfrac{\explain{motion.launch-speed}{v_0}^{2}\sin(2\explain{motion.launch-angle}{\theta})}{\explain{motion.range.gravitational-acceleration}{g}}$

(G.3)

Symbols

$g$gravitational acceleration(meters per second squared)

This formula is correct only if the launch point and the landing point are at the same height and air resistance is zero. The range is symmetric about a launch angle of 45 degrees: two complementary launch angles, whose sum is 90 degrees, give the same range.

Introduced by: motion.compute-range in projectiles.projectile-range

Used directly in: Projectile Range, The Optimal Launch Angle

Source records:

• openstax-university-physics-1 — ch. 4, "Projectile Motion"

$t$

Time

motion.time

draft

Domain

motion

Units

seconds after launch

Elapsed time since the object was launched or released, measured in seconds.

In every scenario of this course, time is zero at the instant the object is launched or released, and time only increases. A scenario has exactly one time origin. So an equation that combines two launches with different time origins contains a modeling error, not a notation error.

Introduced by: motion.describe-position in motion.position-and-coordinates

Used directly in: Position and Coordinate Systems, Projectile Range, Velocity Under Constant Acceleration

Source records:

• openstax-university-physics-1 — ch. 3, "Time, Position, and Displacement"

$\dot{\square}$

Time derivative

motion.time-derivative

draft

Domain

motion

Units

per second

See also

Velocity

The rate of change of a quantity with time, written as a dot over the quantity's letter.

A dot over a letter is the time derivative of that quantity: the dot over the horizontal coordinate is the rate of change of that coordinate with time. Two dots mark the second time derivative. The dot is an operator, so the letter under it keeps its own meaning. The unit of the result is the unit of the quantity divided by seconds.

Introduced by: motion.describe-position in motion.position-and-coordinates

Used directly in: Position and Coordinate Systems

Source records:

• openstax-university-physics-1 — ch. 3, "Instantaneous Velocity and Speed"

$v(t)$

Velocity

motion.velocity

draft

Domain

motion

Units

meters per second

See also

Position

The rate at which position changes with time. It is a vector; in a straight-line problem only its size and sign matter, and that is what the scalar v(t) tracks.

$\explain{motion.velocity}{\mathbf{v}} = \dfrac{d\,\explain{motion.position}{\mathbf{r}}}{d\explain{motion.time}{t}}\,;\quad \explain{motion.velocity}{v(t)} = \explain{motion.launch-speed}{v_0} + \explain{motion.velocity.constant-acceleration}{a}\,\explain{motion.time}{t}$

(G.4)

Symbols

$a$constant acceleration(meters per second squared)

Velocity is the time derivative of Position, and acceleration is the time derivative of velocity. If the acceleration is constant for the whole scenario, integrating the acceleration over time gives the second formula above. This formula is exact, not an approximation. On a straight line, a positive velocity means that the object moves in the positive direction of the axis. A negative velocity means that it moves in the negative direction.

Introduced by: motion.predict-velocity in motion.constant-acceleration

Used directly in: Projectile Range, Velocity Under Constant Acceleration

Source records:

• openstax-university-physics-1 — ch. 3, "Instantaneous Velocity and Speed"

Abbreviations

SI

The International System of Units — the metric system this course's units (meters, seconds, kilograms, joules) are drawn from.