MOCK TEST – 1 (Special Theory of Relativity)

Instructions for the Test

Welcome to the Test! Please read the instructions carefully before starting:

1. Number of Questions: The Test consists of multiple questions. Each question will be displayed one at a time.
2. Answering Questions: For each question, select the correct option from the list of multiple choices. Only one option can be selected per question.
3. Navigation:
   • Click the “Save & Next” button to save your answer and proceed to the next question.
   • You can go back to previous questions by clicking the “Previous” button if needed.
4. Progress Bar: A progress bar will be displayed at the top of the test, showing how many questions you’ve attempted.
5. Review: If you are unsure of an answer, you can navigate back to that question and change your response before submitting the test.
6. Results: After completing all the questions, your score will be displayed at the end of the test. The total number of correct answers will be shown.
7. Timing: There is no time limit for this quiz, so take your time to read and answer each question carefully.

Good luck! Click “Start” to begin the test.

Inertial Frame of Reference

An inertial frame of reference is a frame of reference in which an object either remains at rest or moves at a constant velocity unless acted upon by an external force. This concept stems from Newton’s First Law of Motion (the law of inertia), which states that an object will continue in its state of motion unless influenced by a net external force. Essentially, in an inertial frame, the laws of physics, especially Newton’s laws of motion, hold true without requiring corrections for acceleration or rotation.

Characteristics of an Inertial Frame of Reference:

1. No Acceleration: An inertial frame is either at rest or moving with a constant velocity (i.e., it is not accelerating).
2. Newton’s Laws Hold True: Newton’s laws of motion can be directly applied without modification in an inertial frame.
3. Relative Nature: No absolute inertial frame exists; one frame may be inertial relative to another if they move at constant velocity with respect to each other.

An example of an inertial frame is a train moving at a constant speed on a straight track. If you were inside the train and threw a ball, the motion of the ball would behave as it would if you were standing still on the ground.

Galilean Transformation

The Galilean transformation is a mathematical framework used to relate the coordinates of an event as seen from one inertial frame to those from another inertial frame. It was introduced by Galileo Galilei to explain how the motion of objects is observed in different reference frames moving relative to each other at a constant velocity. The Galilean transformation assumes absolute time and absolute space, meaning time passes at the same rate for all observers, and space remains unaffected by the relative motion of reference frames.

Galilean Transformation Equations

Consider two inertial frames of reference:

• Frame S, which is stationary.
• Frame S′, which moves with a constant velocity V relative to S along the x-axis.

Let t and t′ represent time in frames S and S′, respectively. Similarly, let x, y, z represent the coordinates in frame S, and x′, y′, z′ represent the coordinates in frame S′.

The transformation between the two frames is described by the following equations:

These equations imply:

• x-coordinate transformation: The position in the x-direction depends on the velocity V of the second frame. The moving frame’s position x′ is found by subtracting the product of time t and velocity V from the position x in the stationary frame.
• y and z transformation: The y and z coordinates remain unchanged because the relative motion between frames occurs only in the x-direction.
• Time invariance: The time t is the same in both frames (absolute time), which is a key feature of Galilean transformation.

Velocity Transformation

The Galilean velocity transformation is obtained by differentiating the position transformation equations with respect to time:

where v_x​, v_y​, and v_z​ are the components of velocity in the stationary frame S, and v_x'​, v_y'​, and v_z'​ are the components in the moving frame S′. This shows that the velocity in the x-direction differs by the relative velocity V between the frames, while the y and z components remain unchanged.

Assumptions of Galilean Transformation

1. Time is Absolute: Time is considered to be the same for all observers, regardless of their state of motion.
2. Space is Absolute: The spatial coordinates transform linearly, and distances between objects are preserved.
3. Velocities Add Linearly: The velocity of an object as seen from different frames is related by simple addition or subtraction of the relative velocity of the frames.

Limitations of Galilean Transformation

While the Galilean transformation works well at low velocities (i.e., when objects move much slower than the speed of light), it breaks down at higher velocities approaching the speed of light. This is because the Galilean transformation does not take into account the finite speed of light and the effects of time dilation and length contraction that occur at relativistic speeds.

For high-velocity scenarios, special relativity and the Lorentz transformation must be used instead. Unlike Galilean transformation, Lorentz transformation accounts for the fact that the speed of light is constant in all inertial frames, leading to the notion of time relativity and length relativity.