Motion in a plane

Motion in plane class 11 is part of kinematics which comes under the mechanics branch of physics. You have heard about the long jump and javelin through this all phenomena come under the chapter motion in the plane of concept Projectile Motion. Understanding how objects move in two dimensions, influenced by both horizontal velocity and vertical acceleration due to gravity, is crucial for understanding mechanics in Physics. This chapter serves as the foundation for solving more problems in later chapters, such as dynamics and Work, Energy And Power. Let's read the entire article to know in depth of motion in a plane class 11 chapter.

Motion In A Plane Class 11 Topics (NCERT Syllabus)

The motion of a plane class 11 has a total of ten concepts, which we discuss one by one.

1. Introduction

In the last chapter, we studied distance, displacement, and acceleration which were important in describing Motion In A Straight Line. This chapter involves understanding the motion in two dimensions, such as in horizontal and vertical directions. This chapter on kinematics introduces vector concepts, which are important for describing motion and Forces in different scenarios.

2. Scalars and Vectors

Quantities are classified as scalars and vectors are based on the direction. Scalar is a quantity that has only magnitude for example: distance, mass, time, and temperature. On the other hand, a vector has both magnitude and direction. Vectors are used to represent quantities like displacement, Velocity, force, and acceleration, which cannot be fully described without specifying their direction.

• Difference Between Scalar and Vector
• Position and Displacement Vectors

3. Multiplication of Vectors by Real Numbers

When any vector is multiplied by any real number (scalar), its direction will not change, but its magnitude will change only.

1. If the scalar is positive, the direction of the vector remains the same, and its magnitude increases or decreases depending on the value of the scalar.

2. If the scalar is negative, the direction of the vector is reversed, but its magnitude is still scaled by the absolute value of the scalar.

For example, if you have a vector v and a scalar k, the new vector resulting from multiplying v by k is written as k * v.

- If k = 2, the magnitude of the vector doubles, but the direction remains the same.

- If k = -3, the magnitude of the vector is scaled by a factor of 3, but the direction of the vector is reversed.

• Multiplication Of Vectors

4. Addition and Subtraction of Vectors

In this topic, we study the Vector Addition And Vector Subtraction which is mainly done by the method Triangle Law of Vector Addition and parallelogram law of vector addition. To add vectors, place the tail of one vector at the head of the other, and the resultant vector is drawn from the tail of the first vector to the head of the last. For subtraction, you can reverse the direction of the vector to be subtracted and then apply the head-to-tail method.

5. Resolution of Vectors

Resolution of vector is the process of breaking a vector into horizontal and vertical components, usually along the x-axis and y-axis. This makes vector analysis easier, especially when dealing with motion in a plane, as the motion in each direction can be analyzed independently.

6. Vector Addition — Analytical Method

In the above topic, we have studied about resolution of vectors. The same approach we are applying for the addition of vectors in the analytical method, in which vectors are added by breaking them into their component along the x-axis and y-axis and then later components are added algebraically.

7. Motion in a Plane

Till now we have studied vector and their components, from this topic Motion starts in a plane which is defined as the movement of an object in two dimensions. it involves understanding how objects changes their position along x and y direction.

8. Motion in a Plane with Constant Acceleration

When an object moves in a plane with constant acceleration, its motion can be described using the equations of motion for both the x and y components. These equations help predict the object's position, velocity, and acceleration at any given point in time.

• Derivation of Equation of Motion

9. Relative Velocity in Two Dimensions

Relative velocity is the velocity of one object as observed from another moving object. In two dimensions, relative velocity involves both the magnitude and direction of the velocity difference between the two objects. This concept is very very important for competitive exams like JEE Main, NEET and other state engineering exams.

• Boat River Problem
• Rain-man Problem

10. Projectile Motion

Projectile Motion refers to the motion of an object that is thrown or projected into the air and is influenced by gravity. The object follows a curved trajectory, and its motion is analyzed by separating it into horizontal and vertical components. This is one of the very important concepts of this chapter which involves many formulae and numerical questions.

• Horizontal Projectile Motion
• Projectile On An Inclined Plane

11. Uniform Circular Motion

Now, coming to our last concept of this chapter name Uniform circular motion occurs when an object moves in a circle with constant speed. Even though the speed is constant, the direction of the object’s velocity is continually changing, which means the object experiences centripetal acceleration. This type of motion requires a force (centripetal force) to keep the object moving along the circular path.

• Centripetal and Centrifugal Force
• Derivation of Centripetal Acceleration

Motion in a plane Class 11 Important Formula

Some of the important formulas of motion in plane class 11 are given below:

Motion in a Plane With Constant Acceleration

$\begin{gathered}\vec{v}=\vec{v}_o+\vec{a} t \\ \vec{r}=\vec{r}_o+\vec{v}_o t+\frac{1}{2} \vec{a} t^2 \\ x=x_o+v_{o x} t+\frac{1}{2} a_x t^2 \\ v_x=v_{o x}+a_x t \\ y=y_o+v_{o y} t+\frac{1}{2} a_y t^2 \\ v_y=v_{o y}+a_y t\end{gathered}$

These are the equations of motion in two dimensions,

where:

$\vec{v}$ is the velocity vector, $\vec{r}$ is the position vector, $\vec{d}$ is the acceleration vector, $t$ is time, The subscripts $o$ represent initial values, $\quad x$ and $y$ represent the components along the $x$-axis and $y$-axis, respectively.

Relative Velocity in Two- Dimension

The velocity of object $A$ relative to $B$

$
\bar{V}_{A B}=\bar{V}_A-\bar{V}_B
$

where $\vec{V}_A$ and $\vec{V}_B$ are velocities in the same frame.
Similarly, $\vec{V}_{B A}=\bar{V}_B-\bar{V}_A$

$
\bar{V}_{A B}=-\bar{V}_{B A} \text { and }\left|\bar{V}_{A B}\right|=\left|\bar{V}_{B A}\right|
$

Projectile Motion

• For motion along X-axis,

$v_x=u_x+a_x t$ and $x=x_0+u_x t+\frac{1}{2} a_x t^2$

• For motion along Y-axis,

$v_y=u_y+a_y t$ and $y=y_0+u_y t+\frac{1}{2} a_y t^2$

• Angular projection of projectile :

1. Time of flight ( $T$ ): $\mathrm{T}=\frac{2 u \sin \theta}{g}$
2. Maximum height(h): $h=\frac{u^2 \sin ^2 \theta}{2 g}$
3. Horizontal range (R): $\mathrm{R}=\frac{u^2 \sin 2 \theta}{g}$
4. Maximum horizontal range $\left(\mathrm{R}_{\mathrm{max}}\right)$ : $R_{\max }=\frac{u^2}{g}$ for $\theta=45^{\circ}$

Note: For maximum range, $\theta \text { should be } 45 \text { degrees. }$

Real-Life Example of Motion in a Plane Class 11

• Projectile Motion: A football kicked into the air follows a parabolic trajectory, moving forward (horizontal plane) and upward (vertical plane) simultaneously.
• Circular Motion: A car making a turn on a curved road exhibits motion in a plane, combining linear and angular motion.
• River Crossing: A boat crossing a river moves forward with its engine's thrust while being pushed sideways by the river's current, resulting in a two-dimensional path.

Importance of Motion in a Plane Class 11

From this class 11 physics chapter 4 motion in a Plane, you can only expect one question from the chapter. This chapter lays the groundwork for the next topic, rotation, which is crucial for JEE. As a result, we recommend that you do not skip this chapter. Circular motion will be discussed several times in Mechanics.

Exam-wise Weightage of Motion In A Plane Class 11

Approach to Solve Motion in a Plane Questions

Here are some important steps are given which will help to solve motion in a plane numerical:

Understand the Problem Setup

Identify the objects involved and their initial conditions (e.g., initial velocity, angle of projection). Break the motion into horizontal (x-axis) and vertical (y-axis) components and determine the known values such as initial velocity, angle of projection, and gravitational acceleration.

Resolve the Initial Velocity into Components

If the object is projected at an angle then break into horizontal and vertical components.

Use Kinematic Equations

The two directions (x and y) are treated independently using the basic kinematic equations.

Horizontal motion (no acceleration in the horizontal direction, assuming no air resistance):

$x=v_{0 x} \cdot t$

Vertical motion (under the influence of gravity):

$y=v_{0 y} \cdot t-\frac{1}{2} g t^2$

Find Key Quantities

Depending on the problem, use the formula which values or unknowns you have to find.

Solve for Unknowns

• Use the known quantities to solve for unknowns (e.g., time of flight, range, maximum height).
• Use vector addition if the problem involves multiple objects or additional forces.
• For complex projectile motion problems, you might need to use simultaneous equations to solve for time, displacement, or velocity.

NCERT Solutions Subject-wise link:

• NCERT solutions for class 11 Physics.
• NCERT solutions for class 11 Chemistry.
• NCERT solutions for class 11 Mathematics.
• NCERT solutions for class 11 Biology.