Mechanics is the backbone of physics and a crucial part of the IIT JEE syllabus. By solving numerical problems, you not only strengthen your concepts but also build problem-solving speed and accuracy. In this guide, we explore key areas — kinematics and dynamics — with illustrative problems.

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Kinematics: Motion Fundamentals

Kinematics deals with describing motion in terms of displacement, velocity, acceleration, and time. Let’s look at classic problems that every IIT JEE aspirant must master.

Problem 1: Projectile Motion Fundamentals

A projectile is launched with initial velocity ##v_0## at angle ##\theta## to the horizontal. Breaking the velocity into components:

• Horizontal: ##v_{0x} = v_0 \cos\theta##
• Vertical: ##v_{0y} = v_0 \sin\theta##

Key results:

• Time of Flight: ### T = \frac{2 v_0 \sin\theta}{g} ###
• Range: ### R = \frac{v_0^2 \sin 2\theta}{g} ###
• Maximum Height: ### H = \frac{v_0^2 \sin^2\theta}{2g} ###

These formulas are cornerstones for projectile-based questions in JEE.

Problem 2: The Falling Projectile

A projectile is launched horizontally with velocity ##v_x## from height ##h##.

• Time to hit ground: ### t = \sqrt{\tfrac{2h}{g}} ###
• Horizontal range: ### R = v_x \, t ###

The independence of horizontal and vertical motion is the key principle here.

Dynamics: Applying Newton’s Laws

Dynamics explains why objects move, by relating forces and accelerations. Two classical problems illustrate this.

Problem 3: Block on an Inclined Plane

For a block of mass ##m## on an incline angle ##\theta##:

• Parallel force: ##mg \sin\theta##
• Perpendicular force: ##mg \cos\theta##

If no friction: ### a = g \sin\theta. ###

If friction coefficient = ##\mu##: ### a = g \sin\theta – \mu g \cos\theta. ###

Problem 4: Connected Masses (Pulley System)

Two masses ##m_1## and ##m_2## connected over a pulley:

• Acceleration: ### a = \frac{(m_2 – m_1)g}{m_1 + m_2} ###
• Tension: ### T = m_1(g + a) = m_2(g – a) ###

This classic Atwood machine problem tests Newton’s second law in systems.

Key Takeaways

• Projectile motion combines horizontal uniform motion with vertical accelerated motion.
• Kinematics requires mastering equations of motion, while dynamics applies Newton’s laws to forces.
• Inclined planes and pulley problems are standard JEE favorites for testing concepts of force resolution and system analysis.
• Consistent practice on diverse problems builds both intuition and accuracy.

Summary Table

Concept	Formula	Application
Projectile (Time of Flight)	### T = \frac{2 v_0 \sin\theta}{g} ###	Finding total flight duration
Projectile (Range)	### R = \frac{v_0^2 \sin 2\theta}{g} ###	Finding horizontal distance
Projectile (Max Height)	### H = \frac{v_0^2 \sin^2\theta}{2g} ###	Peak vertical displacement
Horizontal Launch	### t = \sqrt{\tfrac{2h}{g}}, \; R = v_x t ###	Impact time and range
Inclined Plane	### a = g \sin\theta – \mu g \cos\theta ###	Acceleration with/without friction
Pulley (Connected Masses)	### a = \frac{(m_2 – m_1)g}{m_1+m_2} ###	Acceleration & tension in Atwood machine

Similar Problems — Quick Applications in Mechanics

Here are a few solved examples that apply kinematics and dynamics concepts. These reinforce key formulas and prepare you for IIT JEE–style questions.

Problem 1: Vertical Throw

A ball is thrown vertically upwards with initial velocity ##u = 15## m/s. Maximum height is given by:

### H = \frac{u^2}{2g} = \frac{15^2}{2 \times 9.8} = 11.48 \,\text{m}. ###

Answer: 11.48 m

Problem 2: Accelerated Car

A car starts from rest (##u=0##) with acceleration ##a=2## m/s² for ##t=10## s. Distance traveled:

### s = ut + \tfrac{1}{2} a t^2 = 0 + \tfrac{1}{2} \times 2 \times 10^2 = 100 \,\text{m}. ###

Answer: 100 m

Problem 3: Friction on a Block

Block of mass ##m=5## kg, applied force ##F=20## N, coefficient of friction ##\mu=0.2##. Frictional force:

### f = \mu m g = 0.2 \times 5 \times 9.8 = 9.8 \,\text{N}. ###

Net force: ##F_{\text{net}} = 20 – 9.8 = 10.2## N. Acceleration:

### a = \frac{F_{\text{net}}}{m} = \frac{10.2}{5} = 2.04 \,\text{m/s}^2. ###

Answer: 2.04 m/s²

Problem 4: Pulley with Two Masses

Masses ##m_1=2## kg, ##m_2=3## kg. Acceleration of system:

### a = \frac{(m_2 – m_1) g}{m_1 + m_2} = \frac{(3-2)\times 9.8}{5} = 1.96 \,\text{m/s}^2. ###

Answer: 1.96 m/s²

Problem 5: Projectile Range

Projectile with ##v_0 = 30## m/s at angle ##\theta = 30^\circ##. Range:

### R = \frac{v_0^2 \sin 2\theta}{g} = \frac{30^2 \sin 60^\circ}{9.8}. ###

### R = \frac{900 \times 0.866}{9.8} \approx 79.5 \,\text{m}. ###

Answer: 79.5 m

Summary Table

Problem	Scenario	Formula Used	Answer
1	Vertical throw (u=15 m/s)	### H = u^2 / 2g ###	11.48 m
2	Car accelerating from rest (a=2, t=10)	### s = ut + \tfrac{1}{2} a t^2 ###	100 m
3	Block with friction (m=5 kg, F=20 N, μ=0.2)	### a = (F – \mu m g)/m ###	2.04 m/s²
4	Pulley (m1=2 kg, m2=3 kg)	### a = (m_2 – m_1)g/(m_1+m_2) ###	1.96 m/s²
5	Projectile (v=30 m/s, θ=30°)	### R = v^2 \sin 2\theta / g ###	79.5 m

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