Analyzing motion along a line using derivatives:

• The velocity of an object is the derivative of its position
• The acceleration of an object is the second derivative of its position
• What is the relationship between position, velocity, and acceleration?

  Derivative relationships
• What is the formula to find acceleration from velocity?

  $a(t) =$$dv / dt$
• What is the formula for velocity in terms of position?

  $v(t) =$$ds / dt$
• What is the second derivative of position called?

  Acceleration
• The velocity function $v(t) =$$6t - 4$ means the particle's velocity changes over time.


  True
• The acceleration of an object is the derivative of the velocity
• If s(t) = t^3 - 6t^2 + 9t, the velocity is v(t) = 3t^2 - 12t + 9


  True
• What is the relationship between velocity and position?

  Velocity is ds/dt
• What does a positive value of acceleration indicate about an object's velocity?

  Velocity is increasing
• Under what condition is an object considered at rest?

  Velocity is zero
• In which interval is the speed decreasing when v(t) = t^2 - 4t + 3 and a(t) = 2t - 4?

  2 < t < 3
• The derivative of position is velocity
• What is the acceleration if s(t) = t^3 - 6t^2 + 9t?

  6t - 12
• The formula for velocity in terms of position is v(t) = ds/dt
• The acceleration of an object is the derivative of its velocity.


  True
• What does a negative value of acceleration indicate about an object's velocity?

  Velocity is decreasing
• A negative value of acceleration means the object's velocity is decreasing.


  True
• If v(t) = t^2 - 4t + 3, the object is at rest when t = 1 or t = 3.


  True
• Speed increases when v(t) and a(t) have the same sign.


  True
• Steps to determine when an object is stationary, moving forward, or backward using v(t) = t - 2

  1️⃣ Solve v(t) = 0 to find stationary points

  2️⃣ Solve v(t) > 0 to find forward motion

  3️⃣ Solve v(t) < 0 to find backward motion
• What is the formula for calculating displacement?

  Δs = s(b) - s(a)
• Match the concept with its characteristic:

  Displacement ↔️ Positive or negative sign

  Distance Traveled ↔️ Always positive
• What is the average velocity of the object from t = 0 to t = 4 if s(t) = t^3 - 6t^2 + 9t?

  -1
• What is the mathematical relationship between velocity and acceleration?

  $a(t) =$$dv / dt$
• Velocity is found by differentiating the position function.

  True
• The velocity of an object is the rate of change of its position
• The position of an object is defined as its location
• If the position function is s(t) = 3t^2 - 4t + 1</latex>, what is the velocity function?

  $v(t) =$$6t - 4$
• What is the acceleration function if the velocity function is $v(t) =$$3t^{2} - 12t +$$9$?


  $a(t) =$$6t - 12$
• The acceleration of an object is the derivative of its velocity
• The position of an object is represented by s(t), where t is time


  True
• Steps to analyze motion using velocity and acceleration functions

  1️⃣ Find when v(t) = 0 to determine rest points

  2️⃣ Determine when v(t) > 0 or v(t) < 0 to find direction

  3️⃣ Calculate a(t) = dv/dt

  4️⃣ Analyze the signs of v(t) and a(t) to find when speed is increasing or decreasing
• An object is moving forward when its velocity is negative.

  False
• Speed increases when velocity and acceleration have opposite signs.

  False
• At what time is an object stationary if v(t) = t - 2?

  t = 2
• What is the velocity if s(t) = t^3 - 6t^2 + 9t?

  3t^2 - 12t + 9
• What is the velocity if v(t) = ds/dt and s(t) = 3t^2 - 4t + 1?

  v(t) = 6t - 4
• The acceleration is the rate of change of velocity