Frame Of Referencemotion In A Straight Lineuniform Topic

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