Introduction

Motion is a fundamental characteristic of the universe. From the microscopic flow of blood in our veins to the macroscopic movement of galaxies, everything is in a state of motion. Motion is defined as the change in the position of an object with time. The branch of physics that studies motion is called Mechanics, which is further divided into Kinematics and Dynamics.

In this chapter, we will focus on Kinematics, which is the study of motion without considering the forces that cause it. We will learn how to describe motion using concepts like velocity and acceleration. Our study will be confined to motion along a straight line, also known as rectilinear motion.

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The Concept of a Point Object

In many situations, the size of an object is negligible compared to the distance it travels. In such cases, we can approximate the object as a point object or a particle. This simplification allows us to describe its motion without worrying about the complexities of its size, shape, or rotation.

For example, when studying the revolution of the Earth around the Sun, the Earth's diameter is insignificant compared to the radius of its orbit, so it can be treated as a point object.

Position, Path Length and Displacement

Frame of Reference

To describe the position of an object, we need a frame of reference. A frame of reference consists of:

• A reference point or origin (O).
• A set of coordinate axes (e.g., three mutually perpendicular axes X, Y, and Z).
• A clock to measure time.

An object is said to be in motion if one or more of its coordinates change with time. If its coordinates do not change with time, it is said to be at rest with respect to that frame of reference.

For rectilinear motion, we can use a single axis, say the X-axis, which coincides with the object's path. Positions to the right of the origin are taken as positive, and positions to the left are taken as negative.

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Path Length

Path length is the total length of the path traversed by an object during its motion. It is the actual distance covered by the object.

• It is a scalar quantity, meaning it has only magnitude and no direction.
• It can never be negative or zero (if the object has moved).

Example: If a car moves from origin O to point P (+360 m) and then back to point Q (+240 m), the path length is:

$ \text{Path Length} = OP + PQ = 360 \text{ m} + (360 - 240) \text{ m} = 360 \text{ m} + 120 \text{ m} = 480 \text{ m} $

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Displacement

Displacement is defined as the change in the position of an object. It is the shortest distance between the initial and final positions.

• It is a vector quantity, having both magnitude and direction. In one-dimensional motion, direction can be indicated by a positive (+) or negative (-) sign.
• Formula: If $x_1$ and $x_2$ are the initial and final positions at times $t_1$ and $t_2$ respectively, the displacement $\Delta x$ is:

  $ \Delta x = x_2 - x_1 $

Example: Using the same car motion from O to P and then to Q:

The displacement is calculated from the initial position (O, where $x_1=0$) to the final position (Q, where $x_2 = +240$ m).

$ \text{Displacement} = \Delta x = x_2 - x_1 = (+240 \text{ m}) - (0 \text{ m}) = +240 \text{ m} $

Here, the magnitude of displacement (240 m) is not equal to the path length (480 m). The magnitude of displacement is either less than or equal to the path length.

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Position-Time Graph (x-t Graph)

A graph plotted with time on the x-axis and position on the y-axis is called a position-time graph. It is a powerful tool for analyzing motion.

• Object at Rest: The position-time graph is a straight line parallel to the time axis.
• Object in Uniform Motion: The position-time graph is a straight line inclined to the time axis.

Average Velocity and Average Speed

Average Velocity

Average velocity is defined as the ratio of the displacement to the time interval in which the displacement occurs. It tells us how fast, on average, an object's position has changed and in what direction.

• It is a vector quantity.
• Formula: $ \bar{v} = \frac{\text{Displacement}}{\text{Time Interval}} = \frac{x_2 - x_1}{t_2 - t_1} = \frac{\Delta x}{\Delta t} $
• The SI unit is m/s or m s$^{-1}$.
• It can be positive, negative, or zero.
• Graphically, the average velocity is the slope of the secant line connecting the initial and final points on a position-time graph.

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Average Speed

Average speed is defined as the ratio of the total path length travelled to the total time interval.

• It is a scalar quantity.
• Formula: $ \text{Average Speed} = \frac{\text{Total Path Length}}{\text{Total Time Interval}} $
• It is always positive for a moving object.

The magnitude of average velocity is generally less than the average speed. It is equal to the average speed only when the object moves along a straight line in the same direction.

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Instantaneous Velocity and Speed

Instantaneous Velocity

The average velocity gives an overall picture of motion over an interval, but not the velocity at a specific moment. The velocity of an object at a particular instant of time is called its instantaneous velocity (or simply, velocity).

It is defined as the limit of the average velocity as the time interval $\Delta t$ approaches zero.

• Formula (Calculus): $ v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} $
• Instantaneous velocity is the first derivative of position with respect to time.
• Graphically, it is the slope of the tangent to the position-time (x-t) graph at that particular instant.

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Instantaneous Speed

Instantaneous speed, or simply speed, is the magnitude of the instantaneous velocity. For example, velocities of +24.0 m/s and -24.0 m/s both correspond to a speed of 24.0 m/s.

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Acceleration

When the velocity of an object changes with time, the object is said to be accelerating. Acceleration is the rate of change of velocity with time.

Average Acceleration

Average acceleration is the change in velocity divided by the time interval over which the change occurs.

• Formula: $ \bar{a} = \frac{v_2 - v_1}{t_2 - t_1} = \frac{\Delta v}{\Delta t} $
• SI unit is m/s$^2$ or m s$^{-2}$.
• It is a vector quantity.
• Graphically, it is the slope of the secant line on a velocity-time (v-t) graph.

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Instantaneous Acceleration

Instantaneous acceleration is the acceleration of an object at a particular instant of time.

• Formula (Calculus): $ a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt} = \frac{d}{dt}\left(\frac{dx}{dt}\right) = \frac{d^2x}{dt^2} $
• It is the first derivative of velocity with respect to time, or the second derivative of position with respect to time.
• Graphically, it is the slope of the tangent to the velocity-time (v-t) graph at that instant.

Acceleration can be positive, negative, or zero.

• Positive acceleration: Velocity is increasing in the positive direction.
• Negative acceleration (retardation or deceleration): Velocity is decreasing in the positive direction or increasing in the negative direction.
• Zero acceleration: Velocity is constant (uniform motion).

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Area under the Velocity-Time Graph

An important feature of a v-t graph is that the area under the curve represents the displacement of the object over that time interval.

For an object moving with constant velocity $u$, the v-t graph is a horizontal line. The area under the graph from $t=0$ to $t=T$ is the area of a rectangle, which is $u \times T$. This is precisely the displacement ($x = ut$).

Kinematic Equations for Uniformly Accelerated Motion

For motion in a straight line with constant acceleration ($a$), a set of simple equations, known as the equations of motion, can be derived. These equations relate displacement ($x$), time ($t$), initial velocity ($v_0$), final velocity ($v$), and acceleration ($a$).

Derivation of the Equations

Let's assume an object starts at position $x_0=0$ at time $t=0$ with an initial velocity $v_0$. After time $t$, its final velocity is $v$ and its position is $x$.

1. First Equation of Motion: $v = v_0 + at$

By definition, constant acceleration is $a = \frac{v - v_0}{t - 0}$.

$at = v - v_0$

$v = v_0 + at$

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2. Second Equation of Motion: $x = v_0t + \frac{1}{2}at^2$

The displacement $x$ is the area under the velocity-time graph. For uniform acceleration, this graph is a straight line, and the area is a trapezium.

Area = Area of rectangle OACD + Area of triangle ABC

$x = (v_0)(t) + \frac{1}{2}(v - v_0)(t)$

From the first equation, we know $(v - v_0) = at$. Substituting this:

$x = v_0t + \frac{1}{2}(at)(t)$

$x = v_0t + \frac{1}{2}at^2$

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3. Third Equation of Motion: $v^2 = v_0^2 + 2ax$

We can also express the displacement $x$ as the area of the trapezium using the formula for average velocity:

$x = (\text{average velocity}) \times t = \left(\frac{v_0 + v}{2}\right)t$

From the first equation, we can write $t = \frac{v - v_0}{a}$. Substituting this for $t$:

$x = \left(\frac{v + v_0}{2}\right) \left(\frac{v - v_0}{a}\right) = \frac{v^2 - v_0^2}{2a}$

$2ax = v^2 - v_0^2$

$v^2 = v_0^2 + 2ax$

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General Equations of Motion

If the initial position is $x_0$ instead of 0, the equations are modified by replacing $x$ with $(x - x_0)$:

1. $v = v_0 + at$
2. $x = x_0 + v_0t + \frac{1}{2}at^2$
3. $v^2 = v_0^2 + 2a(x - x_0)$

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Relative Velocity

The concept of velocity is relative. The velocity of an object depends on the frame of reference from which it is observed. To understand this, we introduce the concept of relative velocity.

Consider two objects A and B moving along a straight line (x-axis) with uniform velocities $v_A$ and $v_B$ with respect to the ground.

Their positions at time $t$ are:

$x_A(t) = x_A(0) + v_At$

$x_B(t) = x_B(0) + v_Bt$

The displacement from object A to object B at time $t$ is:

$x_{BA}(t) = x_B(t) - x_A(t) = [x_B(0) - x_A(0)] + (v_B - v_A)t$

This equation shows that, as seen from object A, object B's position changes by an amount $(v_B - v_A)$ each second. Therefore, the velocity of B relative to A is:

$v_{BA} = v_B - v_A$

Similarly, the velocity of A relative to B is:

$v_{AB} = v_A - v_B$

From this, we can see that $v_{BA} = -v_{AB}$.

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Special Cases

1. If $v_A = v_B$: Then $v_{BA} = 0$. The two objects remain at a constant distance from each other. To an observer on A, B appears to be at rest. Their position-time graphs are parallel straight lines.
2. If $v_A > v_B$ (moving in same direction): Then $v_{BA}$ is negative. Object A will eventually overtake object B.
3. If $v_A$ and $v_B$ have opposite signs: The magnitude of relative velocity, $|v_B - v_A|$, will be greater than both $|v_A|$ and $|v_B|$. The objects will appear to move very fast with respect to each other.

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Elements of Calculus

Calculus is a powerful branch of mathematics that deals with continuous change. It has two major branches: Differential Calculus and Integral Calculus. These tools are essential for defining and working with physical quantities like instantaneous velocity and acceleration.

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Differential Calculus

Differential calculus is concerned with the rate at which quantities change. The core concept is the derivative.

If a quantity $y$ is a function of another quantity $x$, written as $y = f(x)$, the derivative of $y$ with respect to $x$ (denoted as $dy/dx$) represents the instantaneous rate of change of $y$ with respect to $x$.

Graphical Interpretation of the Derivative

Graphically, the derivative of a function at a point is the slope of the tangent line to the function's graph at that point. It is found by taking the limit of the slope of a secant line between two points as the distance between the points approaches zero.

The derivative is formally defined as the limit:

$ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $

Application in Physics

In kinematics, derivatives are used to define instantaneous velocity and acceleration:

• Instantaneous Velocity ($v$): It is the time derivative of position ($x$).

$ v = \frac{dx}{dt} $

• Instantaneous Acceleration ($a$): It is the time derivative of velocity ($v$), or the second time derivative of position ($x$).

$ a = \frac{dv}{dt} = \frac{d^2x}{dt^2} $

Common Derivatives

Below are some fundamental rules and derivatives of common functions. Here, $u$ and $v$ are functions of $x$, and $a$ and $n$ are constants.

Rules of Differentiation:

• Constant Rule: $\frac{d}{dx}(a) = 0$
• Product Rule: $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$
• Quotient Rule: $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$

Function $f(x)$	$x^n$	$\sin(x)$	$\cos(x)$	$\tan(x)$	$e^x$	$\ln(x)$
Derivative $\frac{df}{dx}$	$nx^{n-1}$	$\cos(x)$	$-\sin(x)$	$\sec^2(x)$	$e^x$	$\frac{1}{x}$

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Integral Calculus

Integral calculus is, in essence, the reverse process of differentiation. It is primarily used to find the area under a curve. This is useful for calculating quantities that are the result of accumulation, such as displacement from velocity or work done by a variable force.

The area under the curve of a function $f(x)$ from $x=a$ to $x=b$ is found by summing the areas of an infinite number of infinitesimally thin rectangles. This limit of a sum is called a definite integral and is denoted by:

$ \text{Area} = \int_{a}^{b} f(x)dx $

The symbol $\int$ is an elongated 'S', signifying a sum.

The Fundamental Theorem of Calculus

This theorem establishes the inverse relationship between differentiation and integration. It states that if the derivative of a function $g(x)$ is $f(x)$, then the definite integral of $f(x)$ from $a$ to $b$ can be easily evaluated:

If $\frac{d}{dx}g(x) = f(x)$, then:

$ \int_{a}^{b} f(x)dx = [g(x)]_{a}^{b} = g(b) - g(a) $

The function $g(x)$ is called the indefinite integral of $f(x)$.

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Common Integrals

Function $f(x)$	$x^n$ (for $n \neq -1$)	$x^{-1}$ or $\frac{1}{x}$	$\sin(x)$	$\cos(x)$	$e^x$
Indefinite Integral $\int f(x)dx$	$\frac{x^{n+1}}{n+1} + C$	$\ln|x| + C$	$-\cos(x) + C$	$\sin(x) + C$	$e^x + C$

Note: The constant 'C' is the constant of integration, which is added because the derivative of any constant is zero. For definite integrals, this constant cancels out.

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Additional Exercises

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