Detailed Notes on Frame of Reference - Motion in a Straight Line - Uniform

Reference Frames

• Inertial Reference Frame: A reference frame in which Newton’s laws of motion hold true.

• Non-Inertial Reference Frame: A reference frame in which Newton’s laws of motion do not hold true, often due to acceleration of the frame.

• Examples:

  • Inertial Reference Frame: Earth’s surface (approximately) for small regions and short time intervals
  • Non-Inertial Reference Frame: Inside a moving car, a rotating turntable

• Relativity of Motion:

  • Motion is relative to the observer’s reference frame.
  • The same object can be at rest in one frame of reference and in motion in another.

Frame of Reference and Laws of Motion

• Newton’s laws of motion are valid only in inertial reference frames.

• In non-inertial frames, additional fictitious forces (e.g., centrifugal force) appear to act on objects.

• Examples:

  • A person inside a rotating car experiences a centrifugal force that pushes them outward.
  • A ball dropped from a tower appears to curve due to Earth’s rotation, creating a fictitious Coriolis force.

Relative Velocity and Motion

• Relative Velocity: The velocity of an object relative to another object.

• Formula for Relative Velocity:

  v_rel = v_obj1 - v_obj2
  

  where:

  • v_rel is the relative velocity
  • v_obj1 is the velocity of object 1
  • v_obj2 is the velocity of object 2

• Examples:

  • A person walking at 5 km/h on a train moving at 10 km/h has a relative velocity of 15 km/h with respect to the ground.
  • Two cars moving in the same direction at 60 km/h and 70 km/h respectively have a relative velocity of 10 km/h.

Galilean Transformations

• Galilean Transformations: Equations that relate the position and velocity of an object in one inertial reference frame to its position and velocity in another inertial reference frame.

• Mathematical Expressions for Galilean Transformations:

  • Position:

    x' = x - vt
    y' = y
    z' = z
    

  • Velocity:

    v_x' = v_x - v
    v_y' = v_y
    v_z' = v_z
    

    where:

    • x, y, z are the coordinates of the object in the original frame
    • t is the time
    • v is the velocity of the new reference frame relative to the original frame
    • x’, y’, z’ are the coordinates of the object in the new frame
    • v_x’, v_y’, v_z’ are the velocity components of the object in the new frame
    • v_x, v_y, v_z are the velocity components of the object in the original frame

Equations of Motion in Uniform Motion

• For an object moving with uniform velocity v in a straight line, the equations of motion are:

  • Position:

    x = x_0 + vt
    

  • Velocity:

    v = constant
    

  where:

  • x is the position of the object at time t
  • x_0 is the initial position of the object
  • v is the velocity of the object

• Examples:

  • A car traveling at 60 km/h covers a distance of 30 km in 30 minutes.
  • A train moving at 100 km/h passes a station in 1 minute.

Graphs and Motion Analysis

• Position-Time Graphs:

  • A plot of the position of an object versus time.
  • Slope of the graph represents the velocity of the object.

• Velocity-Time Graphs:

  • A plot of the velocity of an object versus time.
  • Area under the graph represents the displacement of the object.

• Examples:

  • A position-time graph for an object moving with constant velocity is a straight line with a constant slope.
  • A velocity-time graph for an object moving with constant acceleration is a straight line with a constant slope.

Examples and Problem-Solving

• Solved Examples:

  • A train passes a station at 50 km/h. A passenger on the train throws a ball forward at 10 km/h. What is the velocity of the ball relative to the ground?
  • A car travels for 2 hours at 60 km/h and then for 3 hours at 80 km/h. Calculate the average velocity of the car for the entire journey.

• Practice Problems:

  • A boat is moving upstream at 5 km/h. The river is flowing at 2 km/h. What is the velocity of the boat relative to the shore?
  • A particle moves according to the equation: x = 10t - 2t^2. Find its velocity at time t = 3s.

References

• NCERT Physics Part 1 (Class 11): Chapter 3 - Motion in a Straight Line
• NCERT Physics Part 2 (Class 12): Chapter 5 - Laws of Motion
• NCERT Physics Part 2 (Class 12): Chapter 6 - Motion in a Plane