Kinematics

Kinematics describes how objects move without any reference to force. Focus is mainly on motion in one dimension (x-axis). Two-dimensional motion (e.g., projectile motion) is a topic for another day.

Key Concepts

Scalar vs. Vector Quantities

• Scalar Quantity: Has magnitude only (e.g., mass, distance, temperature).

  • Example: Mass of a book is 2 kg (no direction).
  • Distance is a scalar quantity; it’s always positive.

• Vector Quantity: Has both magnitude and direction (e.g., displacement, velocity, acceleration).

  • Example: Displacement of 15 meters east indicates direction.
  • Displacement is defined as the change in position:
    • Displacement = Final Position - Initial Position.

Speed vs. Velocity

• Speed: A scalar quantity that describes how fast an object is moving.

  • Example: 20 m/s (magnitude only).

• Velocity: A vector quantity that includes speed and direction.

  • Example: 30 m/s north (magnitude + direction).

Temperature

• Temperature is a scalar quantity (magnitude only).
  • Example: 80°F (no direction).

Acceleration

• Acceleration is a vector quantity that indicates how fast velocity changes.
  • Example: 2 m/s² east.

Distance vs. Displacement

• Distance: Total length of the path traveled (scalar, always positive).
• Displacement: Difference between final and initial position (vector).
  • Example: If a person travels 30 m east then 4 m west:
    • Total Distance = 30 m + 4 m = 34 m.
    • Displacement = 30 m (east) - 4 m (west) = 26 m (east).

Average Speed vs. Average Velocity

• Average Speed: Total distance traveled divided by time.
• Average Velocity: Total displacement divided by time.
  • Example Problem:
    • Particle travels 100 m east then 150 m west in 5 seconds:
      • Total Distance = 100 + 150 = 250 m.
      • Average Speed = 250 m / 5 s = 50 m/s.
      • Displacement = 100 m - 150 m = -50 m.
      • Average Velocity = -50 m / 5 s = -10 m/s.

Instantaneous Speed and Velocity

• Instantaneous Speed: Absolute value of instantaneous velocity.
• Instantaneous Velocity: Velocity at a specific moment in time.

Equations of Motion

Constant Speed

• Formula: d = vt
  • d = distance or displacement; v = speed or velocity.

Constant Acceleration

• Displacement Formula: d = v̅t (v̅ = average velocity).
• Average Velocity Formula: v̅ = (v_initial + v_final) / 2.
• Rearranged Equation:
  • x_final = x_initial + v̅t
  • v_final = v_initial + at
  • v_final² = v_initial² + 2ad
  • d = v_initial t + 0.5at²

Example Problem

• Bus traveling at constant speed: 40 m/s.

  • To find time to travel 200 miles:
    • Convert 40 m/s to mph: 40 m/s = 89.5 mph.
    • Use d = vt to find time (t).

• Average Velocity Calculation:

  • Bus travels from 50 miles east to 90 miles west in 5 hours.
  • Displacement = x_final - x_initial = -90 - 50 = -140 miles.
  • Average Velocity = -140 miles / 5 hours = -28 mph.

Recap

• Distance is scalar (always positive). Displacement is vector (can be positive or negative).
• Speed is scalar, while velocity includes direction.
• Remember to convert units when necessary.