5.8 Connecting Position, Velocity, and Acceleration

• In the context of motion, position is denoted by s(t)
• Match the term with its derivative:

  Position ↔️ Velocity

  Velocity ↔️ Acceleration

  Acceleration ↔️ a'(t)
• Velocity indicates how quickly and in what direction an object's position is changing.


  True
• The velocity of an object at time t is denoted by `v(t)`.
• Acceleration tells us how quickly the velocity is changing.


  True
• The derivative of velocity is acceleration, denoted as `v'(t) = a(t)`.


  True
• If `s(t) = 2t^3 - 4t^2 + 3t + 1`, then the velocity is 6t^2 - 8t + 3.
• What is the derivative relationship between position and velocity?

  v(t) = s'(t)
• If `s(t) = 2t^3 - 4t^2 + 3t + 1`, then the velocity is `v(t) = 6t^2 - 8t + 3
• How is velocity calculated from position using derivatives?

  v(t) = s'(t)
• Acceleration is the rate of change of velocity.


  True
• Position, velocity, and acceleration are all defined in terms of time.


  True
• What does velocity tell us about an object's position?

  How quickly it changes
• How is acceleration mathematically related to velocity?

  a(t) = v'(t)
• What is the first step to find velocity from a position function?

  Take the first derivative
• If `s(t) = 3t^2 - 5t + 2`, then the acceleration is `a(t) = 6
• What is the velocity for the position function `s(t) = 2t^3 - 4t^2 + 3t + 1`?

  6t^2 - 8t + 3
• Velocity is the first derivative of position.


  True
• Acceleration is the rate of change of velocity
• Match the term with its definition:

  Position ↔️ Location of object at time t

  Velocity ↔️ Rate of change of position
• The relationship between position and velocity is given by `v(t) = s'(t)`.


  True
• Acceleration is the derivative of velocity with respect to time.
• The derivative of velocity, `v'(t)`, gives the acceleration of the object.
• Match the term with its derivative:

  Position ↔️ Velocity

  Velocity ↔️ Acceleration
• Velocity is the rate of change of position.


  True
• If `s(t) = 2t^3 - 4t^2 + 3t + 1`, then the acceleration is `a(t) = 12t - 8
• How is acceleration calculated from velocity using derivatives?

  a(t) = v'(t)
• What is the derivative of position with respect to time?

  Velocity
• If `s(t) = 2t^2 - 3t + 1`, then the velocity is `v(t) = 4t - 3
• If `v(t) = 3t^2 - 6t + 2`, then the acceleration is `a(t) = 6t - 6
• If `s(t) = 3t^2 - 5t + 2`, then the velocity is `v(t) = 6t - 5`.


  True
• Acceleration is the rate of change of velocity.

  True
• If `s(t) = 2t^3 - 4t^2 + 3t + 1`, then the acceleration is `a(t) = 12t - 8