Detailed Notes on Mechanics for JEE Preparation based on NCERT (Class 11 and 12)

1. Frame of Reference:

• Definition: A frame of reference is a set of objects used to describe the motion of other objects.
• Inertial Frame: A frame of reference in which Newton’s laws of motion hold true is called an inertial frame.
• Non-Inertial Frame: A frame of reference in which Newton’s laws of motion do not hold true is called a non-inertial frame.

2. Motion in a Straight Line:

• Description: Motion in a straight line involves the movement of an object along a straight path.
• Position: The location of an object along a straight line at a given instant of time.
• Displacement: The change in the position of an object during a specific time interval.
• Distance: The total length of the path covered by an object during its motion.
• Speed: The rate of change of distance with respect to time.
• Velocity: The rate of change of displacement with respect to time.

3. Uniform Motion:

• Concept and Definition: Uniform motion is motion with a constant velocity.
• Equations of Uniform Motion:
  • $$s = ut$$
  • $$v = u$$ where $$s$$ is the displacement, $$u$$ is the initial velocity, $$t$$ is the time taken, and $$v$$ is the final velocity.
• Graphical Representation: A position-time graph of uniform motion shows a straight line with a constant slope, while a velocity-time graph of uniform motion shows a horizontal line.

4. Relative Motion:

• Concept and Definition: Relative motion is the motion of an object relative to another object.
• Examples:
  • A person walking on a moving train is in relative motion to the train.
  • A car moving on a road is in relative motion to the ground.
• Relative Velocity: The velocity of an object with respect to another object. It can be calculated by subtracting the velocity of the second object from the velocity of the first object.

5. Laws of Motion:

• Newton’s First Law of Motion (Law of Inertia): An object at rest will remain at rest, and an object in motion will continue to move at a constant velocity in a straight line unless acted upon by an external force.
• Newton’s Second Law of Motion: The acceleration of an object is directly proportional to the net force applied to it, and inversely proportional to its mass.
• Newton’s Third Law of Motion (Principle of Action-Reaction Forces): For every action, there is an equal and opposite reaction.

6. Equations of Motion:

• Derivation: $$v = u + at$$
• Application: Uniformly Accelerated Motion: $$s = ut + \frac{1}{2}at^2$$
• Problem-Solving:
  • Determine the initial velocity, acceleration, or final velocity of an object from its position-time graph or its velocity-time graph.
  • Calculate the displacement or distance traveled by an object during a specific time interval.
  • Derive the equations of motion for uniformly accelerated motion.
  • Solve numerical problems involving uniformly accelerated motion.

7. Motion Under Gravity:

• Concepts:
  • Objects near the Earth’s surface experience a force of gravity that pulls them downwards.
  • Motion under gravity can be analyzed using the equations of motion for uniformly accelerated motion.
• Equations of Motion:
  • $$v = u + gt$$
  • $$s = ut + \frac{1}{2}gt^2$$

8. Projectile Motion:

• Concept and Trajectory: A projectile is a body that is projected into space, usually by being thrown, shot, or dropped. Its motion is called projectile motion. The path that the projectile follows is known as its trajectory.
• Horizontal and Vertical Components: Projectile motion can be resolved into horizontal and vertical components. The horizontal motion is uniform motion, while the vertical motion is motion under gravity.
• Equations of Projectile Motion:
  • Horizontal Motion: $$R = u_x t$$
  • Vertical Motion: $$h = u_y t + \frac{1}{2}gt^2$$ where $$R$$ is the horizontal range, $$u_x$$ is the initial horizontal velocity, $$t$$ is the time of flight, $$h$$ is the maximum height, and $$u_y$$ is the initial vertical velocity.

9. Graphical Representation of Motion:

• Position-Time Graphs: Slope of the graph gives velocity, and the area under the graph gives the displacement.
• Velocity-Time Graphs: Slope of the graph gives acceleration.
• Acceleration-Time Graphs: The area under the graph gives change in velocity.