NotesShala Class 11 · Physics

Chapter02

Motion in a Straight Line

General Introduction of Motion · Uniformly Accelerated Motion

Reading key Definition Note Remember Derivation Strategy Formulas sit on graph paper · coloured words are the key terms

Part 1

General Introduction of Motion

An object is in motion when its position changes with respect to its surroundings. Motion is all around us. Examples: a train running on its tracks, a person walking, an insect crawling and water flowing through a dam.

1.1Motion

Definition

Motion is the change in position of an object with time, measured from a chosen reference point. The reference can also be a group of objects taken together.

Types of Motion

Based on the path followed

Rectilinear Motion

The particle moves along a straight line.

Circular Motion

The particle moves along a circle.

Oscillatory Motion

The particle moves back and forth about a fixed point.

Based on how many coordinates are needed

One-dimensional (1-D)

Only one coordinate is needed to locate the object.

Two-dimensional (2-D)

Two coordinates are needed to locate the object.

Three-dimensional (3-D)

All three coordinates are needed to locate the object.

1.2Point Object

Definition

An object can be treated as a point object when the distance it covers is very large compared with its own size.

Examples

iExample

A train on a journey of hundreds of kilometres.

iiExample

The Earth while it revolves around the Sun.

1.3Scalar and Vector Quantities

Physical quantities are of two types: scalars and vectors.

iScalars

A quantity that has only magnitude (size) is called a scalar quantity, or simply a scalar.

e.g. Mass, length, time, speed, etc.

iiVectors

A quantity that has both magnitude and direction is called a vector quantity, or simply a vector.

e.g. Displacement, velocity, acceleration, etc.

1.4Position

Note

Position, Path Length & Displacement

To say where an object is, we need a reference point, a set of axes and a frame of reference.

Definition

Position is the point where an object is located. It is measured on a coordinate axis marked in units of length. This axis has a positive and a negative direction.

Positions to the right of the origin are positive. Positions to the left are negative

e.g. An object at −1 m is on the negative side of the origin. The minus sign shows the direction. An object at 0 m is exactly at the origin. Positions are always measured from a convenient origin.

Position to the right of the originPositive (+)

Position to the left of the originNegative (−)

Frame of Reference

We usually take three mutually perpendicular axes as a rectangular coordinate system. These are the $x$ , $y$ and $z$ axes. The point where they meet is the origin (O). It is the reference point.

The $x$ , $y$ and $z$ coordinates give the position of the object with respect to these axes. A clock is used to measure time.

Coordinate system+Clock=Frame of Reference

There are two types of frames of reference:

iInertial Frame of Reference

A frame in which Newton's first law of motion holds true.

iiNon-inertial Frame of Reference

A frame in which Newton's first law of motion does not hold.

Note

Accelerated and Unaccelerated Frames

A frame that is accelerating with respect to the Earth is usually non-inertial. A frame that is not accelerating is inertial.
Frames at rest or moving uniformly are inertial. Frames in non-uniform motion are non-inertial.
To use Newton's second law in a non-inertial frame, we add an extra imaginary force. It is called the pseudo force.

1.5Path Length / Distance

Definition

The path length, or distance travelled, is the total length of the path that an object actually covers in a given time interval. Its SI unit is the metre (m).

Example

An object moves 100 m away from the origin $O$ along the $x$ -axis in time $t$ . So its path length is 100 m. It then comes back to the origin in time $t'$ . Now the total path length is:

\text{Path length} = 100\ \text{m} + 100\ \text{m} = 200\ \text{m}

Forward and back: total path length = 100 m + 100 m = 200 m

1.6Displacement

Definition

Displacement is the change in position of an object in a definite direction. It is equal to the final position minus the initial position. It is written as $\Delta x$ .

Displacement

\Delta x = x_2 - x_1

where, $x_1$ = initial position and $x_2$ = final position of the object

Cases

(i) If

x_2 > x_1

\Delta x

is positive

(ii) If

x_2 < x_1

\Delta x

is negative

(iii) If

x_2 = x_1

\Delta x

is zero

So displacement can be positive, negative or zero. Displacement has both magnitude and direction, so it is a vector quantity.

Note

The magnitude of the displacement may or may not be equal to the path length.
For a trip, the displacement can be zero. But the path length of any actual motion is never zero.

Differences between Distance and Displacement

S.No.	Distance	Displacement
1.	The total length of the path covered.	The length of the shortest path from the initial position to the final position.
2.	A scalar quantity.	A vector quantity.
3.	For a moving object it cannot be zero.	It can be zero.
4.	It is never negative.	It can be positive, negative or zero.

1.7Uniform & Non-uniform Motion

Definition

Uniform Motion in a Straight Line

A body moving along a straight line is in uniform motion if it covers equal distances in equal intervals of time.

e.g. A vehicle moving steadily at 10 m/s covers 10 m every secondUniform

Note

In uniform motion along a straight line in one fixed direction, the magnitude of displacement is equal to the distance covered.

Non-uniform Motion

A body is in non-uniform motion if it covers unequal distances in equal intervals of time.

In such motion, the speed of the body, its direction of motion, or both change with time.

e.g. A vehicle whose velocity is different at different instantsNon-uniform

1.8Speed

Definition

The speed of an object is the distance it covers divided by the time taken.

Speed

\text{Speed} = \dfrac{\text{Distance travelled}}{\text{Time taken}}

Nature

Scalar quantity; always positive.

SI Unit

$\text{m s}^{-1}$ (MKS / SI); $\text{cm s}^{-1}$ (CGS).

Dimensions

$[\mathrm{M^0 L^1 T^{-1}}]$

Types of Speed

Uniform Speed

The object covers equal distances in equal intervals of time.

Variable / Non-Uniform Speed

The object covers unequal distances in equal intervals of time. In other words, it takes unequal times to cover equal distances.

1.9Velocity

Definition

The velocity of an object is the rate of change of its position with time.

Velocity

\text{Velocity} = \dfrac{\text{Displacement}}{\text{Time}}

Nature

Vector quantity.

SI Unit

$\text{m s}^{-1}$ (MKS / SI); $\text{cm s}^{-1}$ (CGS).

Dimensions

$[\mathrm{M^0 L^1 T^{-1}}]$

Types of Velocity

Uniform Velocity

The object undergoes equal displacements in equal intervals of time.

Non-uniform Velocity

The object undergoes unequal displacements in equal intervals of time.

Note

Velocity is positive, zero or negative when the displacement is positive, zero or negative, respectively.

1.10Average Speed and Average Velocity

Average Speed

v_{av} = \dfrac{\text{Total distance travelled}}{\text{Total time taken}}

The whole distance covered, divided by the whole time taken.

Average Velocity

v_{av} = \dfrac{\Delta x}{\Delta t}

The displacement $\Delta x$ (change in position) divided by the time interval $\Delta t$ in which it happens.

Remember

Average Velocity vs Average Speed

Average speed = total distance covered ÷ time taken.
Average velocity = net displacement ÷ time taken for that displacement.
Average velocity can be zero, positive or negative. But the average speed of a moving body is always positive.

1.11Instantaneous Speed and Instantaneous Velocity

Definition

Instantaneous Speed

Instantaneous speed is the speed at a particular instant. It is the limit of the average speed as the time interval $\Delta t$ approaches zero. Here $\Delta s$ is the distance covered in the small time $\Delta t$ .

\text{Speed} = \displaystyle\lim_{\Delta t \to 0} \dfrac{\Delta s}{\Delta t} = \dfrac{ds}{dt}

Definition

Instantaneous Velocity

Instantaneous velocity is the velocity at a particular instant. It is the limit of the average velocity as the time interval $\Delta t$ approaches zero.

v_i = \displaystyle\lim_{\Delta t \to 0} \dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}

Note

$\dfrac{dx}{dt}$ is the derivative of position $x$ with respect to time $t$ . Here $dx$ is the tiny displacement in a tiny time $dt$ .
At any instant, the magnitude of the instantaneous velocity is equal to the instantaneous speed.

Graphical Interpretation of Instantaneous Velocity

When the velocity keeps changing, the position–time graph is a curve. The average velocity between $t_1$ and $t_2$ is the slope of the chord $AB$ . The time interval is $\Delta t = t_2 - t_1$ .

Now make $\Delta t$ smaller. The chord becomes $AC$ , then $AD$ . As $\Delta t \to 0$ , point $B$ moves closer and closer to $A$ . The chord then becomes the tangent to the curve at $A$ . So the slope of the tangent at a point gives the instantaneous velocity there.

Slope of chord

AB

= average velocity. The chord (

AC, AD\ldots

) finally becomes the tangent

EF

A

. Its slope = instantaneous velocity

Part 2

Uniformly Accelerated Motion

2.1Acceleration

Definition

Acceleration is the rate of change of velocity with time. It tells us how fast the velocity of an object changes. Acceleration is a vector quantity.

Acceleration

\text{Acceleration} = \dfrac{\text{Change in velocity}}{\text{Time taken}}

Nature

Vector quantity.

SI Unit

$\text{m s}^{-2}$ (SI); $\text{cm s}^{-2}$ (CGS).

Dimensions

$[\mathrm{M^0 L^1 T^{-2}}]$

Note

An object accelerates when its speed changes, when its direction changes, or when both change.

Velocity and acceleration have the same signBody speeds up

Velocity and acceleration have opposite signsBody slows down (retards)

Average Acceleration

Let the velocity of a body be $v_1$ at time $t_1$ and $v_2$ at a later time $t_2$ . Its average acceleration is the change in velocity divided by the time interval:

a_{av} = \dfrac{\Delta v}{\Delta t} = \dfrac{v_2 - v_1}{t_2 - t_1}

So acceleration is the slope of the velocity–time graph.

Slope of secant

PR

= \dfrac{RQ}{PQ} = \dfrac{\Delta v}{\Delta t} = \tan\theta

= average acceleration

Instantaneous Acceleration

Definition

Instantaneous acceleration is the acceleration of a body at a particular instant. It is the limit of the average acceleration as the time interval approaches zero.

a_{inst} = \displaystyle\lim_{\Delta t \to 0} \dfrac{\Delta v}{\Delta t} = \dfrac{dv}{dt}

Here $\dfrac{dv}{dt}$ is the derivative of $v$ with respect to $t$ .

Remember

Differentiate or Integrate?

Since $v = \dfrac{dx}{dt}$ and $a = \dfrac{dv}{dt} = \dfrac{dv}{dx}\cdot\dfrac{dx}{dt}$ :

If the displacement is given and you need velocity and acceleration, differentiate.
If the acceleration is given and you need velocity and displacement, integrate.

2.2Uniform and Non-uniform Acceleration

Uniform Acceleration

The velocity changes by equal amounts in equal intervals of time.

Non-uniform Acceleration

The velocity changes by unequal amounts in equal intervals of time.

2.3Position-Time Graphs for Positive, Negative and Zero Acceleration

The shape of a position–time (distance–time) graph tells us whether the acceleration is positive, negative or zero.

iPositive Acceleration

The object covers more and more distance in each equal time interval. So the graph curves upward.

iiNegative Acceleration

The object covers less and less distance in each equal time interval. So the graph gets flatter with time.

iiiZero Acceleration

The object covers equal distances in equal times. So the graph is a straight line.

2.4Velocity-Time Graphs for an Accelerated Motion

The shape of a velocity–time (v-t) graph shows what kind of acceleration the object has.

Zero Acceleration

With zero acceleration, the velocity stays constant with time. So the graph is a horizontal line.

Note

The area under a v-t graph gives the displacement.

Positive Acceleration

With positive acceleration, the velocity of the object keeps increasing with time.

aZero initial velocity

Starting from rest, the velocity-time graph is a straight line through the origin.

bSome initial velocity

Starting with some velocity, the velocity-time graph is a straight line beginning at P on the velocity axis.

Increasing Acceleration

When the acceleration itself keeps increasing, the velocity rises faster and faster (non-linearly). The graph shows an object starting from rest. The slope of the v-t graph at any point gives the instantaneous acceleration at that point.

Negative Acceleration

With uniform negative acceleration, the velocity of the object decreases steadily (linearly) with time.

aPositive initial velocity

Starting with a positive velocity, the graph is a straight line sloping downwards (negative slope).

bNegative initial velocity

Starting with a negative velocity, the graph is a straight line that begins at P. P is a point of negative velocity below the time axis.

cPositive initial velocity — direction reverses

Starting with a positive velocity, the velocity can fall to zero and then become negative. So the direction of motion reverses at time $t$ . For uniform acceleration, the slope of the graph stays the same throughout.

2.5Kinematic Equations for Uniformly Accelerated Motion

Definition

Uniformly Accelerated Motion

When the velocity of an object changes by the same amount in every unit of time, it has constant acceleration. Such motion is called uniformly accelerated motion.

Setup — Given Information

The object moves along a straight line with constant acceleration $a$ .
Initial velocity $= u$ (at $t = 0$ ). Final velocity $= v$ (after time $t$ ).
The slope of the $v$ - $t$ graph $= a$ (acceleration).
The area under the $v$ - $t$ graph $=$ displacement $x$ .

Velocity-time graph — trapezium

OPQT

(i) Velocity-Time Relation

Derivation

The slope of the velocity-time graph is equal to the acceleration:

$a = \text{Slope of } PQ = \dfrac{QR}{PR} = \dfrac{v-u}{t}$
$\Rightarrow \quad v - u = at$

Velocity-Time Relation

v = u + at

(ii) Position-Time Relation

Derivation

The area under the velocity-time graph is equal to the displacement. For a time interval $t$ :

$x = \text{Area of } \triangle PQR + \text{Area of rectangle } OPRT$
$= \tfrac{1}{2}\,QR \times PR + OP \times PR$
$= \tfrac{1}{2}(v-u)\,t + ut$
$= \tfrac{1}{2}(at)\,t + ut \qquad [\because v - u = at]$
$= \tfrac{1}{2}at^{2} + ut$
$\Rightarrow \quad x = \tfrac{1}{2}at^{2} + ut$

Position-Time Relation

x = ut + \tfrac{1}{2}at^{2}

(iii) Position-Velocity Relation

Derivation

The displacement in time $t$ is also equal to the area of the trapezium $OPQT$ :

$x = \tfrac{1}{2}(OP + QT)\times OT = \tfrac{1}{2}(u+v)\,t \qquad \text{…(i)}$
$\text{From } v - u = at \;\Rightarrow\; t = \dfrac{v-u}{a} \qquad \text{…(ii)}$
$\text{Substituting (ii) in (i): } \; x = \tfrac{1}{2}(u+v)\times\dfrac{v-u}{a} \;\Rightarrow\; v^{2} - u^{2} = 2ax$

Position-Velocity Relation

v^{2} = u^{2} + 2ax

2.6Displacement of the Particle in n^th Second

Concept

Consider an object with initial velocity $u$ and constant acceleration $a$ . To find its displacement during the $n^{th}$ second, take its position at the end of $n$ seconds. Then subtract its position at the end of $(n-1)$ seconds.

Derivation

$s_{n^{th}} = s_n - s_{n-1}$
$= \left[\,un + \tfrac{1}{2}a\,n^{2}\right] - \left[\,u(n-1) + \tfrac{1}{2}a(n-1)^{2}\right]$
$= u + \tfrac{a}{2}\left[\,n^{2} - (n-1)^{2}\right]$
$= u + \tfrac{a}{2}\left(2n - 1\right)$

Displacement in nth Second

s_{n^{th}} = u + \dfrac{a}{2}\left(2n - 1\right)

where, $s_{n^{th}}$ = displacement in the $n^{th}$ second, $u$ = initial velocity, $a$ = constant acceleration

2.7Steps to Solve Problems Based on Accelerated Motion

A simple method to solve problems on accelerated motion:

Problem Solving Strategy

Accelerated Motion

Check the UnitsMake sure all the units match. If distances are in metres, velocities must be in m/s and accelerations in m/s².
List Knowns and UnknownsWrite down all the given quantities. Separately, write the quantities you need to find.
Pick the Right Equation(s)Think about what is happening physically. Then choose the kinematic equation(s) that connect the given quantities to the unknowns.

2.8Non-uniformly Accelerated Motion

Definition

In non-uniformly accelerated motion, the acceleration of a particle is not constant. For example, it may change with time. We describe such one-dimensional motion with these calculus (differential) relations.

Velocity

v = \dfrac{dx}{dt}

Acceleration

a = \dfrac{dv}{dt} = \dfrac{d^{2}x}{dt^{2}}

Velocity–Displacement

v\,dv = a\,dx

Note

These relations come from calculus and work for uniform as well as non-uniform acceleration.

2.9Equations of Motion for an Object Under Gravity

Definition

Motion Under Gravity

A body is said to be in motion under gravity when it is thrown upwards or falls towards the Earth. Whether it moves up or down, its acceleration is the acceleration due to gravity $(g \approx 9.8\ \text{m/s}^2)$ .

So the equations of motion still apply. Just replace the acceleration $a$ by $(\pm g)$ and the displacement by the height $h$ :

Velocity–Time …(i)

v = u + (\pm g)\,t

Position–Time …(ii)

h = ut + \tfrac{1}{2}(\pm g)\,t^{2}

Position–Velocity …(iii)

v^{2} = u^{2} + 2(\pm g)\,h

The sign of $g$ depends on the direction of motion:

Moving upward — gravity opposes the motion

g

is negative

(-g)

Moving downward — gravity acts along the motion

g

is positive

(+g)

Remember

Feeling Weightlessness

Many people think that no force acts on us in free fall, because we cannot feel our weight. In fact, we "feel" our weight through the reaction force from the ground. In free fall, we are pressing on nothing. So there is no reaction force (Newton's third law), and we feel weightless. But gravity still acts on us.

2.10Galileo's Law of Odd Numbers

Law

Galileo was the first to study free fall with careful measurements. He found that the distances covered by a freely falling body in successive equal intervals of time are in the ratio of the odd numbers starting from 1, i.e. $1:3:5:7\,\ldots$

Ratio of Distances in Successive Equal Time Intervals

1 : 3 : 5 : 7 : \;\cdots\; : (2n - 1)

distances covered in the 1st, 2nd, 3rd, … equal time intervals $\tau$ , in the ratio of odd numbers starting from 1

A freely falling body: distances 1 : 3 : 5 : 7 in equal times

\tau

Proof of the Law

Divide the time of free fall into many equal intervals $\tau$ . Then find the distance covered in each interval. Since $u = 0$ :

y = -\tfrac{1}{2}gt^{2}, \qquad \text{where } y_0 = \tfrac{1}{2}g\tau^{2}

$t$	$y$	$\|y\|$ in terms of $y_0 = \tfrac{1}{2}g\tau^{2}$	Distance traversed in successive intervals	Ratio of distances traversed
$0$	$0$	$0$	—	—
$\tau$	$-\tfrac{1}{2}g\tau^{2}$	$y_0$	$y_0$	$1$
$2\tau$	$-4\!\left(\tfrac{1}{2}g\tau^{2}\right)$	$4y_0$	$3y_0$	$3$
$3\tau$	$-9\!\left(\tfrac{1}{2}g\tau^{2}\right)$	$9y_0$	$5y_0$	$5$
$4\tau$	$-16\!\left(\tfrac{1}{2}g\tau^{2}\right)$	$16y_0$	$7y_0$	$7$
$5\tau$	$-25\!\left(\tfrac{1}{2}g\tau^{2}\right)$	$25y_0$	$9y_0$	$9$
$6\tau$	$-36\!\left(\tfrac{1}{2}g\tau^{2}\right)$	$36y_0$	$11y_0$	$11$

Hence Proved

The third column gives the distance fallen $|y|$ in units of $y_0$ . The fourth column gives the distance covered in each successive interval $\tau$ . These distances are in the simple ratio:

y_0 : 3y_0 : 5y_0 : 7y_0 : 9y_0 : 11y_0 \;=\; 1 : 3 : 5 : 7 : 9 : 11

These are the odd numbers starting from 1. Hence proved.

2.11Stopping Distance for a Vehicle

Definition

The stopping distance $(d_s)$ of a vehicle is the distance it travels after the brakes are applied, until it comes to rest.

Note

Stopping distance is very important for road safety. It depends on the speed of the vehicle and how efficient its brakes are.

Derivation

Let the vehicle travel a distance $d_s$ before stopping. Use the equation of motion $v^{2} = u^{2} + 2ax$ with final velocity $v = 0$ and $x = d_s$ :

$0 = u^{2} + 2a\,d_s$
$\Rightarrow \quad d_s = \dfrac{-\,u^{2}}{2a}$

Stopping Distance

d_s = \dfrac{-\,u^{2}}{2a}

where, $u$ = initial speed, $a$ = acceleration (negative when braking, so $d_s$ is positive)

d_s \propto u^{2}

, so doubling the initial speed (with the same braking)

d_s

becomes 4×

2.12Reaction Time

Definition

When a situation needs immediate action, we still take a little time to respond. The time taken to see, think and act is called reaction time.

Example

A driver suddenly sees a child run onto the road. The time that passes before the driver presses the brakes is the driver's reaction time.

Reaction time depends on

iPresence of mind

How alert the person is.

iiComplexity

How complicated the situation is.

Calculation of Reaction Time

A simple experiment measures it. A friend holds a long ruler vertically between your thumb and forefinger. The friend releases it without warning. You catch it as quickly as you can. By the time you react, the ruler has already fallen a distance $x$ .

Ruler-drop test for reaction time

Derivation

Suppose $x = 0.21\ \text{m}$ . Here $u = 0$ and $a = g = 9.8\ \text{m/s}^2$ . Use $x = ut + \tfrac{1}{2}at^{2}$ to find the reaction time $t$ :

$0.21 = 0 + \tfrac{1}{2}\times 9.8 \times t^{2}$
$t = \sqrt{\dfrac{2\times 0.21}{9.8}} \approx 0.21\ \text{s}$

Reaction Time

t = \sqrt{\dfrac{2x}{g}}

where, $x$ = distance the ruler falls before it is caught, $g$ = acceleration due to gravity

End of Chapter 2 · notesshala.in

General Introduction of Motion

1.1Motion

1.2Point Object

1.3Scalar and Vector Quantities

1.4Position

1.5Path Length / Distance

1.6Displacement

1.7Uniform & Non-uniform Motion

1.8Speed

1.9Velocity

1.10Average Speed and Average Velocity

1.11Instantaneous Speed and Instantaneous Velocity

Uniformly Accelerated Motion

2.1Acceleration

2.2Uniform and Non-uniform Acceleration

2.3Position-Time Graphs for Positive, Negative and Zero Acceleration

2.4Velocity-Time Graphs for an Accelerated Motion

2.5Kinematic Equations for Uniformly Accelerated Motion

2.6Displacement of the Particle in nth Second

2.7Steps to Solve Problems Based on Accelerated Motion

2.8Non-uniformly Accelerated Motion

2.9Equations of Motion for an Object Under Gravity

2.10Galileo's Law of Odd Numbers

2.11Stopping Distance for a Vehicle

2.12Reaction Time

2.6Displacement of the Particle in n^th Second