Relative Motion: Frames & Velocity

• Relative motion is the calculation of the motion of an object with regard to some other moving object.

Frames of Reference

• A frame of reference is a coordinate system used to represent and measure properties of an object, such as its position, orientation, and velocity, at different moments in time.
• Inertial Frames: Frames of reference that are not accelerating and Newton's laws of motion hold true in them.
• Non-inertial Frames: Frames of reference that are accelerating and Newton's laws of motion do not hold true in them without modification (e.g., the addition of fictitious forces).

Relative Position

• The position of an object as observed from a particular frame of reference.
• Denoted as ( \vec{r}_{AB} ), representing the position of object A relative to object B.
• Calculated as ( \vec{r}_{AB} = \vec{r}_A - \vec{r}_B ), where ( \vec{r}_A ) and ( \vec{r}_B ) are the positions of A and B relative to a common origin.

Relative Velocity

• The velocity of an object as observed from a particular frame of reference.
• Denoted as ( \vec{v}_{AB} ), signifying the velocity of object A relative to object B.
• Calculated as ( \vec{v}_{AB} = \vec{v}_A - \vec{v}_B ), where ( \vec{v}_A ) and ( \vec{v}_B ) are the velocities of A and B relative to a common origin.

Relative Acceleration

• The acceleration of an object as observed from a particular frame of reference.
• Denoted as ( \vec{a}_{AB} ), indicating the acceleration of object A relative to object B.
• Calculated as ( \vec{a}_{AB} = \vec{a}_A - \vec{a}_B ), where ( \vec{a}_A ) and ( \vec{a}_B ) are the accelerations of A and B relative to a common origin.

Galilean Transformations

• Used to relate the coordinates and velocities of an event as seen in different inertial frames of reference.
• Position transformation: ( \vec{r}' = \vec{r} - \vec{v}t ), where ( \vec{r}' ) is the position in the moving frame, ( \vec{r} ) is the position in the stationary frame, ( \vec{v} ) is the relative velocity between the frames, and ( t ) is time.
• Velocity transformation: ( \vec{v}' = \vec{v} - \vec{u} ), where ( \vec{v}' ) is the velocity in the moving frame, ( \vec{v} ) is the velocity in the stationary frame, and ( \vec{u} ) is the relative velocity between the frames.
• Time is considered absolute, meaning ( t' = t ).
• Valid for speeds much smaller than the speed of light.

Applications

Projectile Motion

• Analyzing the motion of projectiles from different moving platforms.
• Example: A ball thrown upward inside a moving train showing different motion to observers inside versus outside the train.

Navigation

• Used in air and sea navigation to calculate headings and velocities relative to the wind or water current.
• Determining the course of a ship relative to the water and the shore.

Collision Problems

• Analyzing collisions between moving objects from different frames of reference.
• Calculating the velocities of objects after a collision as perceived by different observers.

Doppler Effect

• Involves relative motion between the source and the observer, impacting observed frequency.
• Observed frequency changes based on the relative velocity between the source and the observer.

Examples in 1D

• Two cars moving in the same direction:
  • If car A moves at 50 m/s and car B moves at 30 m/s in the same direction, the relative velocity of car A with respect to car B is 20 m/s, meaning an observer in car B sees car A moving away at 20 m/s.
• Two cars moving in opposite directions:
  • If car A moves at 50 m/s and car B moves at -30 m/s (opposite direction), the relative velocity of car A with respect to car B is 80 m/s, so an observer in car B sees car A approaching at 80 m/s.

Examples in 2D

• Airplane in crosswind:
  • An airplane flying with a certain velocity while experiencing wind from the side. The resultant velocity of the airplane relative to the ground is the vector sum of the airplane's velocity and the wind's velocity.
• Boat crossing a river:
  • A boat attempting to cross a river with a current resulting in the boat's velocity relative to the shore being the vector sum of the boat's velocity in still water and the river's current velocity.

Importance

• Crucial in many areas of physics and engineering, including classical mechanics, fluid dynamics, and special relativity.
• Allows for the simplification of complex problems by choosing an appropriate frame of reference.
• Essential for accurate predictions and measurements in various real-world applications.

Understand relative motion, frames of reference, and relative velocity. Explore inertial and non-inertial frames. Learn to calculate relative positions and velocities in physics.