NCERT Solutions for Class 11 Physics Chapter 4 – Motion in a Plane
Vectors, projectile motion, uniform circular motion formulas, subtopics, and free PDF for Chapter 4.
About Chapter 4 – Motion in a Plane
Chapter 4, Motion in a Plane, extends kinematics from one dimension (Chapter 3) to two dimensions. This chapter is fundamental because most real-world motions – from a ball thrown at an angle to a planet orbiting the Sun – occur in a plane or in three-dimensional space. The chapter begins with a thorough treatment of vectors, covering addition, subtraction, resolution into components, and the laws of vector addition including the triangle law and parallelogram law.
A major topic is projectile motion – the motion of an object launched at an angle to the horizontal, moving under the influence of gravity alone. Students derive expressions for the time of flight, maximum height, horizontal range, and the trajectory equation (which turns out to be a parabola). The concept of splitting motion into independent horizontal (uniform) and vertical (uniformly accelerated) components is a key analytical technique that students must master.
The chapter also covers uniform circular motion – an object moving along a circular path with constant speed. Although the speed is constant, the velocity keeps changing direction, resulting in a centripetal acceleration directed toward the centre. Students learn to derive the expression for centripetal acceleration and understand the role of centripetal force. For board exams, projectile motion and circular motion are the most numerically intensive topics. JEE aspirants should pay special attention to the derivation of range and height formulas, as well as relative velocity in two dimensions, which is an extension of the concept introduced in Chapter 3.
NCERT Solutions for Class 11 Physics Chapter 4 – Free PDF Download
NCERT Solutions – Chapter 4: Motion in a Plane (All Exercises)
Download solved PDF with all exercise solutions covering vector operations, projectile motion numericals, and circular motion problems.
Important Formulas – Chapter 4: Motion in a Plane
| Concept | Formula | Description |
|---|---|---|
| Resultant of Two Vectors | R = √(A² + B² + 2AB cosθ) | Magnitude of resultant; θ is the angle between A and B |
| Direction of Resultant | tan φ = B sinθ / (A + B cosθ) | Angle φ that resultant makes with vector A |
| Unit Vector | â = A⃗ / |A| | Vector of magnitude 1 in direction of A |
| Dot Product | A⃗ · B⃗ = AB cosθ | Scalar product; used in work calculations |
| Cross Product (magnitude) | |A⃗ × B⃗| = AB sinθ | Vector product; direction given by right-hand rule |
| Projectile – Time of Flight | T = 2u sinθ / g | Total time from launch to landing (same level) |
| Projectile – Maximum Height | H = u² sin²θ / (2g) | Maximum vertical rise above launch point |
| Projectile – Horizontal Range | R = u² sin 2θ / g | Horizontal distance; maximum when θ = 45° |
| Projectile – Trajectory | y = x tanθ − gx²/(2u²cos²θ) | Parabolic path equation |
| Centripetal Acceleration | a_c = v²/r = ω²r | Directed toward centre; for uniform circular motion |
| Angular Velocity | ω = Δθ/Δt = 2π/T = 2πf | Rate of change of angle; rad/s |
| Centripetal Force | F_c = mv²/r = mω²r | Net inward force required for circular motion |
| Relative Velocity (2D) | v⃗_AB = v⃗_A − v⃗_B | Velocity of A relative to B in vector form |
Subtopics of Chapter 4 – Motion in a Plane
4.1 Scalars and Vectors
Scalars have magnitude only (speed, mass); vectors have both magnitude and direction (velocity, force). Vectors follow different addition rules.
4.2 Vector Addition – Graphical Methods
Triangle law and parallelogram law for adding vectors. The resultant is the closing side of the triangle or diagonal of the parallelogram.
4.3 Resolution of Vectors
Any vector can be resolved into two perpendicular components: A_x = A cosθ (horizontal), A_y = A sinθ (vertical).
4.4 Vector Addition – Analytical Method
Add x-components and y-components separately. R_x = A_x + B_x; R_y = A_y + B_y; |R| = √(R_x² + R_y²).
4.5 Motion in a Plane
Position vector r⃗ = xî + yĵ. Velocity and acceleration are also vectors. Equations of motion apply separately to x and y components.
4.6 Projectile Motion
Horizontal: uniform velocity; vertical: uniform acceleration due to gravity. The path is a parabola. Key angle for max range = 45°.
4.7 Uniform Circular Motion
Speed constant but velocity direction continuously changes. Centripetal acceleration = v²/r directed toward centre of circle.
4.8 Relative Velocity in 2D
For two objects moving in a plane, relative velocity is found by vector subtraction. Important for river-boat and rain-umbrella problems.
| Resource Name | Description | Best For |
|---|---|---|
| NCERT Solutions | Detailed answers and explanations for NCERT textbook questions across all classes and subjects. | Homework, assignments, and exam preparation |
| NCERT Solutions for Class 11 | Chapter-wise solutions for all Class 11 subjects including Physics, Chemistry, Mathematics, Biology, and English. | Class 11 board exam preparation |
| NCERT Solutions for Class 11 Physics | Step-by-step solutions covering all chapters such as Motion, Laws of Motion, Work Energy and Power, Thermodynamics, and Waves. | Concept building and numerical problem-solving |
| NCERT Exemplar Class 11 Physics | Advanced and application-based questions designed to strengthen conceptual understanding and analytical skills. | JEE, NEET, Olympiads, and higher-order practice |
| Physics Formula | Chapter-wise collection of important formulas, equations, and derivations for quick revision. | Last-minute revision and numerical practice |
Quick Reference – Projectile Motion Summary
| Parameter | Formula | Value at Special Point |
|---|---|---|
| Initial horizontal velocity | u_x = u cosθ | Same throughout flight (constant) |
| Initial vertical velocity | u_y = u sinθ | Zero at maximum height |
| Vertical velocity at time t | v_y = u sinθ − gt | v_y = 0 at H_max |
| Time to reach max height | t = u sinθ / g | Half the total time of flight |
| Max height H | u² sin²θ / 2g | Maximum when θ = 90° |
| Range R | u² sin2θ / g | Maximum when θ = 45°; R(θ) = R(90°−θ) |
| Speed at any time t | v = √(u_x² + v_y²) | Minimum at top = u cosθ |
| Angle of velocity at t | tan α = v_y / u_x | α = 0 at highest point |
NCERT Solutions for Class 11 Physics Chapter 4: Motion in a Plane – FAQs
Scalar quantities have magnitude only, while vector quantities possess both magnitude and direction. Examples of scalars include mass and temperature, whereas displacement and velocity are vectors. This distinction is crucial because the entire chapter is based on vector analysis. Students learn vector addition, subtraction, and resolution. Understanding vectors helps explain motion in two dimensions and prepares students for projectile motion and rotational mechanics. A strong grasp of scalar and vector quantities is essential for solving numerical problems accurately.
Projectile motion refers to the motion of an object thrown into the air under the influence of gravity. The object simultaneously undergoes horizontal and vertical motion. Students study concepts such as time of flight, maximum height, and horizontal range. Projectile motion is frequently tested because it combines multiple concepts from kinematics. Understanding the independence of horizontal and vertical motions simplifies problem-solving. Regular practice helps students solve projectile motion numericals efficiently and understand real-world applications such as sports and ballistics.
Uniform circular motion occurs when an object moves along a circular path with constant speed. Although speed remains constant, velocity changes continuously because direction changes. This results in centripetal acceleration directed toward the center of the circle. Students often find this concept challenging because acceleration exists even when speed is constant. Understanding uniform circular motion is essential for later chapters involving forces and rotational dynamics. Many practical examples, such as satellites and vehicles moving on curved roads, illustrate this concept.
Relative velocity describes the velocity of one object as observed from another moving object. It helps analyze situations involving moving trains, boats, and aircraft. Questions based on relative velocity are common because they require conceptual understanding rather than direct formula use. Students learn how motion appears different from different frames of reference. Mastering this topic enhances problem-solving skills and develops a deeper understanding of motion in multiple dimensions.
Vector resolution involves breaking a vector into perpendicular components. This method simplifies calculations involving motion in two dimensions. Students frequently use horizontal and vertical components to solve projectile motion and relative velocity problems. Without vector resolution, many calculations become difficult and time-consuming. Understanding this concept improves analytical skills and allows students to solve complex questions systematically. It is also a fundamental tool used throughout higher-level physics.