Glossary

Acceleration

Criticality: 3

Acceleration is the rate of change of velocity with respect to time, describing how quickly an object's velocity is changing. It is the derivative of the velocity function, $a(t) = v'(t)$, or the second derivative of the position function, $a(t) = s''(t)$.

Example:

If a car's acceleration is $a(t) = 6t$ m/s $^2$ , its rate of change of velocity is increasing linearly over time.

Arc Length (for vector-valued functions)

Criticality: 2

Arc length for vector-valued functions is the total length of the curve traced by the function over a specified interval. It is calculated by integrating the magnitude of the derivative of the vector-valued function, $S=\int_a^b |\mathbf r'(t)|dt$.

Example:

To find the total path length of a particle moving along $\mathbf{r}(t) = \langle t^2, t^3 angle$ from $t=0$ to $t=1$ , you would calculate its arc length.

Displacement

Criticality: 3

Displacement is the net change in an object's position from its initial to its final location, regardless of the path taken. It can be found by calculating the definite integral of the velocity function over a given time interval.

Example:

If you walk 5 miles east and then 5 miles west, your displacement is 0 miles, as you returned to your starting point.

Distance Traveled

Criticality: 3

Distance traveled is the total path length covered by an object over a given time interval, accounting for the entire journey regardless of direction. It is calculated by integrating the magnitude (speed) of the velocity function over the interval.

Example:

If you run 2 miles around a circular track and end up back at your starting point, your distance traveled is 2 miles.

Parametric Arc Length

Criticality: 2

Parametric arc length is the total length of a curve defined by parametric functions over a given interval. It is calculated using the formula $S=\int_a^b \sqrt{\left( \frac{dx}{dt} ight)^2 + \left( \frac{dy}{dt} ight)^2} dt$.

Example:

To determine the total distance a robot travels along a path defined by $x(t) = t^3$ and $y(t) = t^2$ from $t=0$ to $t=2$ , you would use the parametric arc length formula.

Parametric Functions

Criticality: 3

Parametric functions define the coordinates of a point (e.g., x and y) as separate functions of a common independent variable, or parameter, typically time ($t$), in the form $x(t)=f(t)$ and $y(t)=g(t)$.

Example:

The motion of a Ferris wheel car can be described by parametric functions like $x(t) = R \cos(t)$ and $y(t) = R \sin(t)$ , where $R$ is the radius.

Position

Criticality: 3

Position refers to the location of an object at any given time within a coordinate system, often denoted as $s(t)$ for one-dimensional motion or $\mathbf{r}(t)$ for multi-dimensional motion.

Example:

If a car's position is given by $s(t) = t^2 + 3t$ , at $t=2$ seconds, its location is $s(2) = 10$ meters from the origin.

Vector-Valued Function

Criticality: 3

A vector-valued function maps a real number (typically time, $t$) to a vector, often representing the position of an object in a coordinate system, such as $\mathbf{r}(t) = \langle f(t), g(t) angle$.

Example:

The path of a projectile can be described by the vector-valued function $\mathbf{r}(t) = \langle 3t, -4.9t^2 + 10t angle$ , where $t$ is time.

Velocity

Criticality: 3

Velocity is the rate of change of position with respect to time, indicating both the speed and direction of an object's motion. It is the first derivative of the position function, $v(t) = s'(t)$.

Example:

If a rocket's velocity is $v(t) = 5t^2$ m/s, at $t=3$ seconds, it's moving at $v(3) = 45$ m/s in the positive direction.

Glossary

Acceleration

Criticality: 3

Example:

If a car's acceleration is $a(t) = 6t$ m/s $^2$ , its rate of change of velocity is increasing linearly over time.

Arc Length (for vector-valued functions)

Criticality: 2

Example:

To find the total path length of a particle moving along $\mathbf{r}(t) = \langle t^2, t^3 angle$ from $t=0$ to $t=1$ , you would calculate its arc length.

Displacement

Criticality: 3

Example:

If you walk 5 miles east and then 5 miles west, your displacement is 0 miles, as you returned to your starting point.

Distance Traveled

Criticality: 3

Example:

If you run 2 miles around a circular track and end up back at your starting point, your distance traveled is 2 miles.

Parametric Arc Length

Criticality: 2

Example:

To determine the total distance a robot travels along a path defined by $x(t) = t^3$ and $y(t) = t^2$ from $t=0$ to $t=2$ , you would use the parametric arc length formula.

Parametric Functions

Criticality: 3

Example:

The motion of a Ferris wheel car can be described by parametric functions like $x(t) = R \cos(t)$ and $y(t) = R \sin(t)$ , where $R$ is the radius.

Position

Criticality: 3

Position refers to the location of an object at any given time within a coordinate system, often denoted as $s(t)$ for one-dimensional motion or $\mathbf{r}(t)$ for multi-dimensional motion.

Example:

If a car's position is given by $s(t) = t^2 + 3t$ , at $t=2$ seconds, its location is $s(2) = 10$ meters from the origin.

Vector-Valued Function

Criticality: 3

A vector-valued function maps a real number (typically time, $t$) to a vector, often representing the position of an object in a coordinate system, such as $\mathbf{r}(t) = \langle f(t), g(t) angle$.

Example:

The path of a projectile can be described by the vector-valued function $\mathbf{r}(t) = \langle 3t, -4.9t^2 + 10t angle$ , where $t$ is time.

Velocity

Criticality: 3

Velocity is the rate of change of position with respect to time, indicating both the speed and direction of an object's motion. It is the first derivative of the position function, $v(t) = s'(t)$.

Example:

If a rocket's velocity is $v(t) = 5t^2$ m/s, at $t=3$ seconds, it's moving at $v(3) = 45$ m/s in the positive direction.