Frame Of Referencemotion In A Straight Lineuniform Topic
Detailed Notes on Frame of Reference - Motion in a Straight Line - Uniform
Reference Frames
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Inertial Reference Frame: A reference frame in which Newton’s laws of motion hold true.
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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.
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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
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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
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Newton’s laws of motion are valid only in inertial reference frames.
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In non-inertial frames, additional fictitious forces (e.g., centrifugal force) appear to act on objects.
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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
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Relative Velocity: The velocity of an object relative to another object.
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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
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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
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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.
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Mathematical Expressions for Galilean Transformations:
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Position:
x' = x - vt y' = y z' = z
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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
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Equations of Motion in Uniform Motion
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For an object moving with uniform velocity v in a straight line, the equations of motion are:
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Position:
x = x_0 + vt
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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
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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
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Position-Time Graphs:
- A plot of the position of an object versus time.
- Slope of the graph represents the velocity of the object.
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Velocity-Time Graphs:
- A plot of the velocity of an object versus time.
- Area under the graph represents the displacement of the object.
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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
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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.
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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