Lecture on Relative Velocity

Introduction to Relative Velocity

• Definition: Relative velocity is the velocity of one object as observed from another object.
• Example Setup: Two cars (A and B) each moving at 75 miles per hour in the same direction are used to illustrate the concept.

Calculating Relative Velocity

• Same Direction Example:

  • Car A and Car B are both moving at 75 mph.
  • Observer (O) watches both cars.
  • Relative Velocity of A to B: 0 mph (since both have the same speed and direction).
  • Velocity of A relative to O: 75 mph.
  • Velocity of B relative to O: 75 mph.

• Opposite Direction Example:

  • If Car B moves in the opposite direction, the relative velocity changes.
  • Formula: V_AB = V_A - V_B.
  • For car A moving at 75 mph east and car B moving 75 mph west:
    • Relative Velocity of A to B = V_A - (-V_B) = 75 - (-75) = 150 mph.

Mathematical Background

• Position Vectors:

  • R_A/O: Position vector of Car A with respect to observer O.
  • R_B/O: Position vector of Car B with respect to observer O.
  • Relative Position Vector R_AB: R_A/O - R_B/O (Position of A with respect to B).

• Derivation of Velocity Formula:

  • Taking the time derivative of the relative position vector gives the relative velocity.
  • Velocity of A relative to B: d/dt (R_A/O - R_B/O) = V_A/O - V_B/O.

Conclusion

• Formula for relative velocity between two objects:
  • V_AB = V_A/O - V_B/O.
• Key Insight: Relative velocity is the difference of the velocities of two objects as measured from a common observer.

Thank you for attending the lecture.