NCERT Class 11 Physics Chapter 3 Notes Motion in a straight Line - Download PDF

NCERT Notes for Class 11 Physics Chapter 2: Download PDF

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NCERT Notes for Class 11 Physics Chapter 2

The NCERT Class 11 Physics Chapter 2 Notes Motion in a Straight Line provide brief explanations and focusing on the understanding of the main concepts such as displacement, velocity, acceleration, and Equation of Motion. These notes are ideal to study quickly and also vital in studying for examinations like CBSE, JEE and NEET.

Rest and Motion

An object is said to be at rest if it remains stationary in relation to a specific frame of reference over time. On the other hand, if an object's position relative to a frame of reference changes over time, it is said to be in motion. A coordinate system to which observers attach coordinates to describe events and observations is referred to as a frame of reference.

Frame of Reference

• A frame of reference is a coordinate system used to describe the position and motion of objects by providing a set of axes relative to an observer's perspective.
• According to the frame of reference:

• A body is considered in motion if its position changes over time relative to that frame, and it is considered at rest if there is no change in its position within that frame of reference. For instance, when observing a moving vehicle from an external reference frame, it appears to be in motion, while from an internal frame within the vehicle, the surroundings may seem stationary.

Motion Along a Straight Line

• A straight-line motion can be effectively described using only the X-axis of a coordinate system. One-dimensional motion is the movement of a body in a straight line that occurs when only one coordinate of the body's position changes with time.Examples of one-dimensional motion include the motion of a car on a straight road and the motion of a freely falling body.

Path Length (Distance) Vs. Displacement

• Path Length: The distance between two points along a route, a scalar quantity that represents the total length travelled.
• Displacement: The change in position of a body, often denoted by ∆x = (x₂ - x₁), and it is a vector quantity indicating the overall change in position. In short, path length considers the total distance travelled, whereas displacement considers the net change in position from the initial to the final point while taking direction into account.

• The magnitude of displacement may or may not be equal to the length of the path.

• When an object returns to its original position along a path with a non-zero length, displacement can be zero. Displacement takes into account the change in position regardless of the total distance travelled, as well as the direction of motion.

Speed

• Speed is defined as the rate of distance covered with time. Here are some characteristics of speed:
• It is a scalar quantity denoted by the symbol v.
• Dimension: [M^oL¹T^-1]
• Unit: Meter/second (S.I.), cm/second (C.G.S.)
• Types of speed

(a) Uniform speed: A particle moves at uniform speed when it covers equal distances in equal intervals of time, regardless of how small those intervals are. For example, a car travels an equal distance of 5 metres per second, indicating a uniform speed of 5 m/s. Uniform speed denotes a constant rate of motion with no acceleration or deceleration during the time intervals specified.

(b) Non-uniform (variable) speed: A particle with non-uniform (variable) speed travels unequal distances in equal time intervals. For example, a car travels 5m in the first second, 8m in the second second, 10m in the third second, 4m in the fourth second, and so on. This variation in distance covered indicates that the particle's speed varies for each one-second interval, confirming that it moves at a variable speed. Variable speed denotes that the rate of motion varies over time.

(c) Average speed: The average speed (V_avg) of a particle for a given interval of time is defined as the ratio of the total distance travelled (d) to the total time taken (t).

$v_{\text {avg }}=\frac{\text { Distance traveled }}{\text { Time taken }}=\frac{d}{t}$

(d) Instantaneous speed: It is the speed of a particle at a specific point in time. When we talk about "speed," we usually mean instantaneous speed.

$$
\text { Instantaneous speed } v=\lim _{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}=\frac{d s}{d t}
$$

Velocity

• Velocity is defined as the rate of change of position or the rate of displacement with time.
• It is a vector quantity having the symbol v.
• Dimension of velocity: [M⁰L¹T^-1]
• Unit: Meter/second (S.I.), cm/second (C.G.S.)
• Types of Velocity

(1) Uniform velocity: The condition in which both the magnitude and direction of an object's velocity remain constant is referred to as uniform velocity. This occurs when the particle continues to move in the same straight line without changing direction. To put it another way, a particle with uniform velocity must travel at a constant speed along a straight path with no change in motion.

(2) Non-uniform velocity: Changes in the magnitude or direction of the velocity, or both, characterise non-uniform velocity. The particle's speed and/or direction of motion may change in this scenario. Non-uniform velocity indicates that the object is not moving in a straight line at a constant speed, but rather has variations in its motion over time.

(c) Average velocity: It is defined as the ratio of the body's displacement to the time it takes.

$$
\text { Averagevelocity }=\frac{\text { Displacement }}{\text { Time taken }} ; \vec{v}_{a v}=\frac{\Delta \vec{r}}{\Delta t}
$$

(d) Instantaneous velocity: Instantaneous velocity is defined as the rate of change of the position vector of a particle with respect to time at a specific instant.

$$
\text { Instantaneous velocity } \vec{v}=\lim _{t \rightarrow 0} \frac{\Delta \vec{r}}{\Delta t}=\frac{d \vec{r}}{d t}
$$

Instantaneous Velocity and Speed

In real-life motion, the speed or velocity of an object may not remain constant throughout its journey. To understand how fast an object is moving at a particular instant of time, we define the concepts of instantaneous speed and instantaneous velocity.

Instantaneous Speed

Instantaneous speed is the speed of an object at a particular moment of time. It is the limit of average speed as the time interval approaches zero. In simpler terms, it tells us how fast an object is moving at an instant.

$
\text { Instantaneous Speed }=\lim _{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}
$

where $\Delta s$ is the small displacement in time interval $\Delta t$.
Instantaneous speed is a scalar quantity and is always positive.

Instantaneous Velocity

Instantaneous velocity is defined as the velocity of an object at a particular instant of time. Like instantaneous speed, it is obtained by taking the limit of the average velocity as the time interval tends to zero.

$
\vec{v}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \vec{x}}{\Delta t}=\frac{d \vec{x}}{d t}
$

where $\vec{x}$ is the displacement vector and $t$ is time.
Instantaneous velocity is a vector quantity and has both magnitude and direction.

Acceleration

• Acceleration is defined as the time rate at which an object's velocity changes. it tells us how quickly and in which direction an object's velocity is changing. It is expressed in acceleration units such as metres per second squared.
• It is a vector quantity with the same direction as the change in velocity (not the velocity itself).
• Dimension of acceleration: $\left[\mathrm{M}^0 \mathrm{~L}^1 \mathrm{~T}^{-2}\right]$
• Units: $\left[\right.$ meter $(\mathrm{m}) /$ second $\left.^2\left(\mathrm{~s}^2\right)\right]$ in $($ S.I. $),\left[\right.$ centimeter $(\mathrm{cm}) /$ second $\left.^2\left(\mathrm{~s}^2\right)\right]$ in (C.G.S.).
• Types of Acceleration

(a) Uniform acceleration: Uniform acceleration refers to a situation in which both the magnitude and direction of the acceleration of a body remain constant during its motion.

(b) Non-uniform acceleration: A body is said to have non-uniform acceleration if there are changes in either the magnitude or direction of acceleration, or both, during its motion.

(c) Average acceleration: The average acceleration of an object is defined as the change in velocity per unit time.

$\vec{a}_{a v}=\frac{\Delta \vec{v}}{\Delta \vec{t}}=\frac{\vec{v}_2-\vec{v}_1}{\Delta t}$

Position-Time, Velocity-Time, and Acceleration-Time Graph

p-t graph

v-t graph

Kinematic Equations for Uniformly Accelerated Motion

Using the velocity-time graph of the object under uniformly accelerated motion, we can derive some simple set of equations that relate displacement $(x)$, time $(t)$, initial velocity $(u)$, final velocity $(v)$ and acceleration (a). Let's discuss the above velocity-time graph of object in detail. The velocity of object changes from $u$ to $v$ in time $t$.

1. Derivation of velocity-time relation

The slope of the graph gives the acceleration a of the object.

$$
\begin{aligned}
& & a & =\frac{B C}{A C}=\frac{v-u}{t} \\
\Rightarrow & & a t & =v-u \\
\Rightarrow & & v & =u+a t ----(i)
\end{aligned}
$$

2. Derivation of displacement-time relation

The area under this curve is
Area of $\triangle A B C+$ area of rectangle $O A C D$
Area of $\triangle A B C=\frac{1}{2} \times(v-u) t$
Area of rectangle $O A C D=$ ut
and as explained in the previous section, the area under $v$ - $t$ curve represents the displacement. Therefore, the displacement $s$ of the object is

$
s=\frac{1}{2}(v-u) t+u t
$-----(ii)

But $(v-u)=a t \quad$ [from equation (i)]
Substituting this value in the above equation, we get
or $\quad s=u t+\frac{1}{2} a t^2$

Derivation of velocity-displacement relation
Now, the area of trapezium $O A B D=\frac{1}{2}$ (sum of parallel sides) $\times$ perpendicular distance between the parallel sides.

$
\begin{aligned}
& S=\frac{1}{2} \times(O A+B D) \times O D \\
& S=\frac{1}{2}(u+v) \times t
\end{aligned}
$

From equation (i), we get

$
t=\frac{v-u}{a}
$

Substituting this value of $t$ in the equation, we get

$
\begin{array}{ll}
& s=\frac{1}{2}(u+v) \times \frac{(v-u)}{a} \\
\Rightarrow & s=\frac{v^2-u^2}{2 a} \\
\Rightarrow & v^2-u^2=2 a s
\end{array}
$

$s$ is the displacement of object in time interval $t$ and is given by change in the position from $x_0$ to $x$

$
s=\left(x-x_0\right)
$

So the three kinematic equations of uniformly accelerated motion that we have derived so far are

$\begin{aligned} & v=u+a t \\ & x-x_0=u t+\frac{1}{2} a t^2 \\ & v^2-u^2=2 a\left(x-x_0\right)\end{aligned}$

Relative Velocity

The relative velocity of an object $A$ with respect to another object $B$ is the velocity with which $A$ appears to move when seen from $B$.

if
$\vec{v}_A$ is the velocity of object A , and
$\vec{v}_B$ is the velocity of object B ,
then the relative velocity of $A$ with respect to $B$ is given by:

$
\vec{v}_{A B}=\vec{v}_A-\vec{v}_B
$

Similarly, the relative velocity of $B$ with respect to $A$ is:

$
\vec{v}_{B A}=\vec{v}_B-\vec{v}_A=-\vec{v}_{A B}
$

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