Motion - Physics - Lucent For GK - UPSC PDF Download

Motion

Speed

Speed is the rate of change of distance with respect to time. It is a scalar quantity and gives how fast an object is moving regardless of its direction.

• Mathematical expression: speed = distance / time.
• SI unit: metre per second (m·s⁻¹). Other common units: km·h⁻¹, cm·s⁻¹.
• Average speed = total distance travelled / total time taken.
• Instantaneous speed is the speed of an object at a particular instant.

Velocity

Velocity is the rate of change of displacement with respect to time. It is a vector quantity: it specifies both magnitude (speed) and direction.

• Mathematical expression for average velocity: velocity = displacement / time.
• Instantaneous velocity is the limit of average velocity as the time interval tends to zero.
• Sign convention: positive or negative sign denotes direction along a chosen axis.

Distance-Time Graphs

• In a distance-time graph, time is plotted on the x-axis and distance on the y-axis.
• For motion with uniform speed, the distance-time graph is a straight line whose slope is constant. Equal distances are covered in equal time intervals.
• The slope of a distance-time graph gives the speed. A steeper straight line indicates a higher speed.

Distance-time graph for an object with non-uniform speed

• For non-uniform speed, the distance-time graph is a curve. If the curve becomes steeper with time, the object is speeding up; if it becomes less steep, the object is slowing down.
• A horizontal line indicates the object is at rest (zero speed).

Acceleration

Acceleration is the rate of change of velocity with respect to time.

• Mathematical expression: acceleration = change in velocity / time interval.
• SI unit: metre per second squared (m·s⁻²).
• Uniform (constant) acceleration means velocity changes by equal amounts in equal time intervals.
• Acceleration due to gravity is the acceleration of a freely falling object near the Earth's surface. It is approximately g ≈ 9.8 m·s⁻² and, for many elementary problems, treated as constant over the Earth's surface.

Velocity-Time Graphs

Velocity-time graph for object moving with uniform accelerations

• In a velocity-time graph, time is on the x-axis and velocity on the y-axis.
• For motion with uniform acceleration, the velocity-time graph is a straight line. The slope of this line equals the acceleration.
• For non-uniform acceleration, the velocity-time graph is a curve. A curve concave upwards typically means acceleration is increasing; concave downwards means acceleration is decreasing.
• The area under a velocity-time graph between two times gives the displacement during that interval.

Velocity-time graph for uniform motion of a car

Velocity time graphs in non-uniformly accelerated motion

Circular motion

When a particle moves with constant speed along a circular path about a fixed point (the centre), the motion is called uniform circular motion. Although the speed is constant, the direction of velocity changes continuously.

Centripetal force and centripetal acceleration

• Because the direction of velocity changes, there must be an acceleration directed towards the centre of the circle; this is the centripetal acceleration.
• For a body of mass m moving with speed v on a circular path of radius r, the centripetal acceleration is a_c = v² / r.
• The required force that produces this acceleration is the centripetal force, with magnitude F = m v² / r, always directed towards the centre of the circle.

Centripetal Force

Newton's Laws of Motion

First law of motion (Law of inertia)

Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise. This statement is known as Newton's first law and is closely related to the concept of inertia.

Inertia

Inertia is the property of matter that resists any change in its state of motion or rest unless acted upon by an external force.

Types of inertia

• Inertia of rest: tendency of a body to remain at rest unless acted on by a force.
• Inertia of motion: tendency of a moving body to continue moving with the same speed along the same straight line unless acted on by a force.
• Inertia of direction: tendency of a body to continue moving in the same direction unless a force changes its direction.

Inertia at rest

Inertia of motion

Examples illustrating the first law

• Why do dry leaves and fruits fall when we shake a tree? When branches are shaken, the branches move but the leaves and fruits tend to remain in their state (inertia). The relative motion causes them to detach and fall.
• Why do passengers fall forward when a bus suddenly stops? Passengers were moving with the bus; when the bus stops, their bodies tend to continue moving forward due to inertia, which makes them lurch forward.

Second law of motion

Newton's second law of motion states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the same direction as the applied force.

• Momentum is the product of mass and velocity: p = m × v.
• For a constant mass, Newton's second law is commonly written as F = m × a, where F is the net force and a the acceleration.
• Impulse is defined as the change in momentum produced by a force acting over a time interval.
• Mathematically, Impulse = Force × Time = change in momentum.

Second Law of Motion

Impulse and practical implication

• A large force acting for a short duration can produce the same change in momentum as a smaller force acting for a longer time.
• This explains the use of cushioned surfaces and airbags in vehicles: increasing the time of impact reduces the force and hence reduces injury for a given change in momentum.

Example related to second law

• Apple falling from a tree: The apple accelerates downward under the force of gravity; the change in its velocity arises because a force (gravity) acts on it, producing acceleration equal to g (acceleration due to gravity).

Third law of motion

To every action there is an equal and opposite reaction.

• Inflated balloon: Air rushing out of an open balloon is the action; the balloon moving in the opposite direction is the reaction.
• Rocket propulsion: Hot gases expelled backwards act as the action; the rocket experiences an equal and opposite reaction forward, causing acceleration of the rocket.

Working Principle of rocket

Common Forces in Mechanics

Friction

Friction is the resistive force that opposes the relative motion (or tendency of motion) between two contacting surfaces or layers of fluid.

• Friction acts tangentially to the surface and opposes motion or impending motion.
• Types of friction include static friction (resists the start of motion) and kinetic (sliding) friction (acts during relative motion).
• Friction depends on the nature of surfaces in contact and the normal force; it does not depend directly on the contact area in the simple models used at this level.
• Practical applications: friction helps walking, braking, and is a source of energy loss in machines (wear and heating).

Friction

Working of Friction