Equations of Motion Lecture Notes

Key Concepts

• Equations of Motion: Determine if the object is moving at constant speed or with constant acceleration.

Constant Speed

• Key Equation: ( d = vt )

  • ( d ) can represent distance or displacement.
  • Distance: Rate multiplied by time or speed multiplied by time.
  • Displacement: Velocity multiplied by time.
  • Rearranging the Equation:
    • Speed: ( v = \frac{d}{t} )
    • Velocity: ( v = \frac{\text{displacement}}{t} )

• Definitions:

  • Velocity: Vector (magnitude and direction).
  • Speed: Scalar (only magnitude, always positive).
  • Distance: Scalar (always positive).
  • Displacement: Vector (can be positive or negative).

• Example: If traveling 8 meters east and 3 meters west:

  • Total Distance: 11 meters.
  • Displacement: ( +8 - 3 = +5 ) meters (net change in position).

Constant Acceleration

• Velocity: Rate of change of displacement.
• Acceleration: Rate of change of velocity.
  • Average Acceleration: ( a = \frac{\Delta v}{\Delta t} = \frac{v_{f} - v_{i}}{t} )
  • Rearranged equation: ( v_{f} = v_{i} + at )

Important Formulas

• Final Velocity: ( v_{f}^{2} = v_{i}^{2} + 2ad )

• Displacement with Constant Acceleration:

  • ( d = \frac{v_{i} + v_{f}}{2} \times t )
  • ( d = v_{i}t + \frac{1}{2}at^{2} )

• Position Formulas (for x and y directions):

  • Final Position: ( x_{f} = x_{i} + v_{i}t + \frac{1}{2}at^{2} )
  • Final Position (y): ( y_{f} = y_{i} + v_{i}t + \frac{1}{2}at^{2} )

Calculus Connections

• Position ( s ) and Derivatives:

  • Velocity: Derivative of position function.
  • Acceleration: Derivative of velocity function.

• Antiderivatives:

  • Velocity from Acceleration: Integral of acceleration plus constant ( c ).
  • Position from Velocity: Integral of velocity plus constant ( c ).

Additional Resources

• Example problems and further explanation on kinematics, projectile motion, free fall problems can be found in the provided links in the description section.