5.8 Connecting Position, Velocity, and Acceleration

• The velocity function is defined as the derivative of the position
• The acceleration function is defined as the derivative of the velocity
• Position, velocity, and acceleration are related through differentiation.
• Arrange the relationship between position, velocity, and acceleration in terms of differentiation.

  1️⃣ Velocity is the first derivative of position

  2️⃣ Acceleration is the first derivative of velocity

  3️⃣ Acceleration is the second derivative of position
• The position function, denoted by $s(t)$, describes the object's location
• The position function $s(t)$ provides both the distance and direction of the object from its reference point.
• Match the function with its notation:

  Position ↔️ $s(t)$

  Velocity ↔️ $v(t) =$$s'(t)$

  Acceleration ↔️ $a(t) =$$v'(t) =$$s''(t)$
• The velocity function $v(t)$ is the derivative of the position
• The speed of an object is the absolute value of its velocity
• Speed only considers the magnitude of velocity without regard to direction.
• Match the function with its relationship to other functions:

  Position ↔️ Given

  Velocity ↔️ First derivative of position

  Acceleration ↔️ Second derivative of position
• The velocity function $v(t)$ is the derivative of the position
• A positive velocity indicates movement in the positive direction, while a negative velocity indicates movement in the negative direction.
• The speed of an object is the absolute value of its velocity
• Speed only considers the magnitude of velocity without regard to direction.
• The velocity function $v(t)$ is the derivative of the position
• The speed of an object is the absolute value of its velocity
• Speed only considers the magnitude of velocity without regard to direction.
• The acceleration function $a(t)$ is the derivative of the velocity
• When the acceleration $a(t) =$$0$, the velocity is constant.
• What does the acceleration function, $a(t)$, represent mathematically?


  Derivative of velocity function
• a(t) = v'(t) = s''(t)
• A negative acceleration indicates that the velocity is decreasing.
• What does the position function, $s(t)$, represent?


  Location of object at time $t$
• v(t) = s'(t)
• What is the relationship between velocity and acceleration?

  Acceleration is derivative of velocity
• The velocity function $v(t)$ is the rate of change of position.
• What does the position function, s(t)</latex>, describe about an object's motion?

  Object's location over time
• v(t) = s'(t)
• A positive velocity indicates movement in the positive direction.
• How is speed calculated from velocity?

  Absolute value of velocity
• v(t) = s'(t)
• Speed refers to the magnitude of velocity without regard to direction.
• What is the mathematical relationship between velocity and position?

  Velocity is derivative of position
• a(t) = v'(t) = s''(t)
• When acceleration is zero, the velocity is constant.
• What is the relationship between position, velocity, and acceleration in calculus?

  Interrelated through differentiation
• The velocity function is the first derivative of the position function.
• What does the second part of the Fundamental Theorem of Calculus allow us to calculate?

  Definite integrals
• The Fundamental Theorem of Calculus connects differentiation and integration