Introduction

In the previous chapter, we described motion along a straight line (one dimension). The directional aspect was handled using positive (+) and negative (-) signs. However, to describe motion in two dimensions (a plane) or three dimensions (space), we need a more robust mathematical tool. This tool is the vector.

This chapter introduces the language of vectors, including their properties and operations like addition, subtraction, and multiplication. We will then use vectors to define physical quantities such as position, displacement, velocity, and acceleration in a plane. We will apply these concepts to analyze two important types of motion in a plane: projectile motion and uniform circular motion.

Scalars and Vectors

Physical quantities can be broadly classified into two categories: scalars and vectors.

Scalars

A scalar quantity is a quantity that has only magnitude and no direction. It is completely specified by a single number along with its proper unit. Scalars are combined using the rules of ordinary algebra.

Examples: Distance, mass, temperature, time, speed, volume, density.

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Vectors

A vector quantity is a quantity that has both magnitude and direction and obeys the triangle law or parallelogram law of vector addition.

Examples: Displacement, velocity, acceleration, force, momentum.

Vector Notation

• In print, vectors are represented by boldface letters (e.g., v, A, r).
• When written by hand, an arrow is placed over the letter (e.g., $\vec{v}, \vec{A}, \vec{r}$).
• The magnitude (or absolute value) of a vector A is a scalar and is written as |A| or simply A.

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Position and Displacement Vectors

Position Vector

To specify the position of an object P in a plane, we choose an origin O. The straight line vector OP, with its tail at the origin O and its head at the point P, is called the position vector of the object. It is usually denoted by r.

Displacement Vector

If an object moves from an initial position P (with position vector r) to a final position P' (with position vector r'), the displacement vector is the vector PP'. It is the change in the position vector.

$\Delta \textbf{r} = \textbf{r'} - \textbf{r}$

The displacement vector is the straight line connecting the initial and final points and its direction points from the initial to the final point. Its magnitude is always less than or equal to the path length.

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Equality of Vectors

Two vectors A and B are considered equal if and only if they have the same magnitude and the same direction. A vector remains unchanged if it is displaced parallel to itself.

Multiplication of Vectors by Real Numbers

Multiplying a vector A by a real number (scalar) $\lambda$ results in a new vector, $\lambda\textbf{A}$.

• The magnitude of the new vector is $|\lambda|$ times the magnitude of the original vector: $|\lambda\textbf{A}| = |\lambda| |\textbf{A}|$.
• If $\lambda$ is positive, the direction of $\lambda\textbf{A}$ is the same as the direction of A.
• If $\lambda$ is negative, the direction of $\lambda\textbf{A}$ is opposite to the direction of A.

For example, the vector $2\textbf{A}$ has twice the magnitude of A and the same direction. The vector $-\textbf{A}$ has the same magnitude as A but the opposite direction.

If the scalar $\lambda$ has dimensions, the resulting vector will have new dimensions. For example, multiplying velocity (a vector) by time (a scalar) gives displacement (a vector).

Addition and Subtraction of Vectors — Graphical Method

Vectors do not add like ordinary numbers; they add geometrically according to specific laws.

Triangle Law of Vector Addition (Head-to-Tail Method)

To add two vectors A and B graphically:

1. Draw vector A.
2. Place the tail of vector B at the head of vector A.
3. The resultant vector R = A + B is the vector drawn from the tail of A to the head of B.

Properties of Vector Addition

• Commutative Law: The order of addition does not matter. A + B = B + A.
• Associative Law: When adding three or more vectors, the grouping does not matter. (A + B) + C = A + (B + C).

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Null Vector (Zero Vector)

A null vector, denoted by 0, is a vector with zero magnitude. Since its magnitude is zero, its direction is indeterminate.

• Adding a vector and its negative gives a null vector: $\textbf{A} + (-\textbf{A}) = \textbf{0}$.
• Adding a null vector to any vector leaves it unchanged: $\textbf{A} + \textbf{0} = \textbf{A}$.
• Multiplying a vector by the number 0 gives a null vector: $0 \times \textbf{A} = \textbf{0}$.

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Subtraction of Vectors

The subtraction of vector B from vector A is defined as the addition of vector A and the negative of vector B.

$\textbf{A} - \textbf{B} = \textbf{A} + (-\textbf{B})$

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Parallelogram Law of Vector Addition

This is an equivalent method to the triangle law for adding two vectors.

1. Draw vectors A and B with their tails at a common origin O.
2. Complete the parallelogram using A and B as adjacent sides.
3. The resultant vector R = A + B is the diagonal of the parallelogram starting from the common origin O.

Resolution of Vectors

Resolution of a vector is the process of splitting a single vector into two or more vectors in different directions, which when added together give the original vector. The split vectors are called the components of the original vector.

Unit Vectors

A unit vector is a vector of unit magnitude (magnitude = 1) and points in a specific direction. It is dimensionless and unitless, used only to specify direction.

• Unit vectors along the x, y, and z-axes of a rectangular coordinate system are denoted by $\hat{\textbf{i}}$, $\hat{\textbf{j}}$, and $\hat{\textbf{k}}$, respectively.
• $|\hat{\textbf{i}}| = |\hat{\textbf{j}}| = |\hat{\textbf{k}}| = 1$.
• Any vector A can be written as the product of its magnitude and a unit vector $\hat{\textbf{n}}$ in its direction: $\textbf{A} = |\textbf{A}|\hat{\textbf{n}}$.

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Rectangular Components of a Vector

It is convenient to resolve a vector along the axes of a rectangular coordinate system. A vector A in the x-y plane can be resolved into two perpendicular components, one along the x-axis ($\textbf{A}_x$) and one along the y-axis ($\textbf{A}_y$).

$\textbf{A} = \textbf{A}_x + \textbf{A}_y$

In terms of unit vectors, this is written as:

$\textbf{A} = A_x\hat{\textbf{i}} + A_y\hat{\textbf{j}}$

Here, $A_x$ and $A_y$ are scalar quantities called the x and y-components of vector A.

Finding Components from Magnitude and Direction

If a vector A has magnitude A and makes an angle $\theta$ with the positive x-axis:

$ A_x = A \cos \theta $

$ A_y = A \sin \theta $

Finding Magnitude and Direction from Components

If the components $A_x$ and $A_y$ are known:

Magnitude: $ A = \sqrt{A_x^2 + A_y^2} $

Direction: $ \theta = \tan^{-1}\left(\frac{A_y}{A_x}\right) $

This concept can be extended to three dimensions, where a vector A is expressed as $\textbf{A} = A_x\hat{\textbf{i}} + A_y\hat{\textbf{j}} + A_z\hat{\textbf{k}}$, and its magnitude is $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.

Vector Addition — Analytical Method

Adding vectors graphically can be tedious. A more precise and easier method is the analytical method, which involves adding the components of the vectors.

Consider two vectors $\textbf{A} = A_x\hat{\textbf{i}} + A_y\hat{\textbf{j}}$ and $\textbf{B} = B_x\hat{\textbf{i}} + B_y\hat{\textbf{j}}$. Their sum, the resultant vector $\textbf{R} = R_x\hat{\textbf{i}} + R_y\hat{\textbf{j}}$, is found by adding their corresponding components:

$ \textbf{R} = \textbf{A} + \textbf{B} = (A_x + B_x)\hat{\textbf{i}} + (A_y + B_y)\hat{\textbf{j}} $

So, the components of the resultant are:

$ R_x = A_x + B_x $

$ R_y = A_y + B_y $

The magnitude and direction of the resultant R can then be found using $R = \sqrt{R_x^2 + R_y^2}$ and $\theta = \tan^{-1}(R_y/R_x)$.

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Law of Cosines and Law of Sines

For two vectors A and B with an angle $\theta$ between them, the magnitude and direction of their resultant R can be found without resolving into components.

Law of Cosines (Magnitude of Resultant)

The magnitude $R$ of the resultant vector $\textbf{R} = \textbf{A} + \textbf{B}$ is given by:

$ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} $

Law of Sines (Direction of Resultant)

If the resultant R makes an angle $\alpha$ with vector A, then:

$ \frac{R}{\sin \theta} = \frac{A}{\sin \beta} = \frac{B}{\sin \alpha} $

Where $\beta$ is the angle between R and B. The direction $\alpha$ can also be found using:

$ \tan \alpha = \frac{B \sin \theta}{A + B \cos \theta} $

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Motion in a Plane

We can now use vectors to describe motion in two dimensions (a plane).

Position Vector and Displacement

The position of a particle P in an x-y plane is described by its position vector r:

$\textbf{r} = x\hat{\textbf{i}} + y\hat{\textbf{j}}$

The displacement $\Delta \textbf{r}$ of the particle as it moves from P to P' is:

$\Delta \textbf{r} = \textbf{r'} - \textbf{r} = (x' - x)\hat{\textbf{i}} + (y' - y)\hat{\textbf{j}} = \Delta x\hat{\textbf{i}} + \Delta y\hat{\textbf{j}}$

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Velocity

The average velocity $\bar{\textbf{v}}$ is the displacement divided by the time interval:

$\bar{\textbf{v}} = \frac{\Delta \textbf{r}}{\Delta t} = \frac{\Delta x}{\Delta t}\hat{\textbf{i}} + \frac{\Delta y}{\Delta t}\hat{\textbf{j}} = \bar{v}_x\hat{\textbf{i}} + \bar{v}_y\hat{\textbf{j}}$

The instantaneous velocity v is the time derivative of the position vector. Its direction is always tangent to the path of the particle.

$\textbf{v} = \lim_{\Delta t \to 0} \frac{\Delta \textbf{r}}{\Delta t} = \frac{d\textbf{r}}{dt}$

In component form:

$\textbf{v} = \frac{dx}{dt}\hat{\textbf{i}} + \frac{dy}{dt}\hat{\textbf{j}} = v_x\hat{\textbf{i}} + v_y\hat{\textbf{j}}$

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Acceleration

The average acceleration $\bar{\textbf{a}}$ is the change in velocity divided by the time interval:

$\bar{\textbf{a}} = \frac{\Delta \textbf{v}}{\Delta t} = \frac{\Delta v_x}{\Delta t}\hat{\textbf{i}} + \frac{\Delta v_y}{\Delta t}\hat{\textbf{j}} = \bar{a}_x\hat{\textbf{i}} + \bar{a}_y\hat{\textbf{j}}$

The instantaneous acceleration a is the time derivative of the velocity vector.

$\textbf{a} = \lim_{\Delta t \to 0} \frac{\Delta \textbf{v}}{\Delta t} = \frac{d\textbf{v}}{dt}$

In component form:

$\textbf{a} = \frac{dv_x}{dt}\hat{\textbf{i}} + \frac{dv_y}{dt}\hat{\textbf{j}} = a_x\hat{\textbf{i}} + a_y\hat{\textbf{j}}$

An important point for 2D/3D motion is that the acceleration vector can have any angle with the velocity vector. For example, in circular motion, they are perpendicular.

Motion in a Plane with Constant Acceleration

If an object moves in a plane with a constant acceleration a, we can derive kinematic equations similar to those for one-dimensional motion.

Let the initial velocity at $t=0$ be $\textbf{v}_0$ and the velocity at time $t$ be v.

From the definition of constant acceleration, $\textbf{a} = (\textbf{v} - \textbf{v}_0)/t$, we get the first equation:

$\textbf{v} = \textbf{v}_0 + \textbf{a}t$

Let the initial position be $\textbf{r}_0$ and the position at time $t$ be r. The displacement $\textbf{r} - \textbf{r}_0$ is the average velocity multiplied by time.

$\textbf{r} - \textbf{r}_0 = \left(\frac{\textbf{v}_0 + \textbf{v}}{2}\right)t = \left(\frac{\textbf{v}_0 + (\textbf{v}_0 + \textbf{a}t)}{2}\right)t$

This simplifies to the second equation:

$\textbf{r} = \textbf{r}_0 + \textbf{v}_0t + \frac{1}{2}\textbf{a}t^2$

A key takeaway from these vector equations is that motion in a plane can be treated as two separate, simultaneous one-dimensional motions along perpendicular axes. The x-component of motion is independent of the y-component of motion.

Relative Velocity in Two Dimensions

The concept of relative velocity extends to two dimensions. If object A moves with velocity $\textbf{v}_A$ and object B moves with velocity $\textbf{v}_B$ (both with respect to a common reference frame, like the ground), then:

The velocity of object A relative to object B is:

$\textbf{v}_{AB} = \textbf{v}_A - \textbf{v}_B$

The velocity of object B relative to object A is:

$\textbf{v}_{BA} = \textbf{v}_B - \textbf{v}_A$

Note that $\textbf{v}_{AB} = -\textbf{v}_{BA}$.

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Projectile Motion

An object that is thrown or projected into the air and is then subject only to the acceleration of gravity is called a projectile. We will assume that air resistance is negligible.

Projectile motion is a classic example of 2D motion with constant acceleration. It can be analyzed as the result of two independent motions:

1. Horizontal Motion: Uniform velocity (zero acceleration).
2. Vertical Motion: Constant downward acceleration ($g$).

Equations of Projectile Motion

Consider a projectile launched from the origin with an initial velocity $v_0$ at an angle $\theta_0$ with the horizontal.

Initial velocity components:

$v_{0x} = v_0 \cos \theta_0$

$v_{0y} = v_0 \sin \theta_0$

Acceleration components:

$a_x = 0$

$a_y = -g$

Velocity at time t

$v_x(t) = v_{0x} = v_0 \cos \theta_0$ (remains constant)

$v_y(t) = v_{0y} + a_y t = v_0 \sin \theta_0 - gt$

Position at time t

$x(t) = v_{0x}t = (v_0 \cos \theta_0)t$

$y(t) = v_{0y}t + \frac{1}{2}a_y t^2 = (v_0 \sin \theta_0)t - \frac{1}{2}gt^2$

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Key Characteristics of Projectile Motion

Equation of Path (Trajectory)

By eliminating time $t$ from the position equations, we get the equation for the path:

$y = (\tan \theta_0)x - \left(\frac{g}{2v_0^2 \cos^2 \theta_0}\right)x^2$

This is the equation of a parabola.

Time of Maximum Height ($t_m$)

At the maximum height, the vertical component of velocity is zero ($v_y = 0$).

$0 = v_0 \sin \theta_0 - gt_m$

$t_m = \frac{v_0 \sin \theta_0}{g}$

Maximum Height ($h_m$)

Substitute $t_m$ into the equation for y(t).

$h_m = \frac{(v_0 \sin \theta_0)^2}{2g}$

Time of Flight ($T_f$)

This is the total time the projectile is in the air. Due to symmetry, $T_f = 2t_m$.

$T_f = \frac{2v_0 \sin \theta_0}{g}$

Horizontal Range (R)

This is the horizontal distance covered during the time of flight.

$R = v_x T_f = (v_0 \cos \theta_0)\left(\frac{2v_0 \sin \theta_0}{g}\right) = \frac{v_0^2(2 \sin \theta_0 \cos \theta_0)}{g}$

Using the identity $\sin 2\theta = 2\sin\theta\cos\theta$:

$R = \frac{v_0^2 \sin(2\theta_0)}{g}$

The range is maximum when $\sin(2\theta_0)$ is maximum (i.e., 1), which occurs when $2\theta_0 = 90^\circ$, or $\theta_0 = 45^\circ$.

Uniform Circular Motion

When an object moves along a circular path at a constant speed, its motion is called uniform circular motion. Although the speed is constant, the velocity is continuously changing because its direction is changing at every point. Since the velocity changes, the object is accelerating.

Centripetal Acceleration ($a_c$)

For an object in uniform circular motion, the acceleration is always directed towards the center of the circle. This acceleration is called centripetal acceleration (meaning "center-seeking").

Derivation of Magnitude

Consider an object moving with speed $v$ in a circle of radius $R$. In a small time interval $\Delta t$, it moves from position P to P' and its position vector rotates by an angle $\Delta \theta$. Its velocity vector also rotates by the same angle $\Delta \theta$.

Using similar triangles formed by the position vectors ($\textbf{r}, \textbf{r'}$) and the velocity vectors ($\textbf{v}, \textbf{v'}$), we can show:

$\frac{|\Delta \textbf{v}|}{v} = \frac{|\Delta \textbf{r}|}{R}$

Since $|\Delta \textbf{v}| = a \Delta t$ and for small intervals $|\Delta \textbf{r}| \approx v \Delta t$, we have:

$\frac{a \Delta t}{v} = \frac{v \Delta t}{R}$

This gives the magnitude of the centripetal acceleration as:

$a_c = \frac{v^2}{R}$

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Angular Speed, Period, and Frequency

Angular Speed ($\omega$): The rate of change of angular position. $\omega = \frac{\Delta \theta}{\Delta t}$. Its unit is rad/s.

Relation between Linear and Angular Speed:

$v = R\omega$

Using this, the centripetal acceleration can also be expressed as:

$a_c = \frac{(R\omega)^2}{R} = \omega^2 R$

Time Period (T): The time taken to complete one revolution. The distance covered is $2\pi R$.

$v = \frac{2\pi R}{T}$

Frequency (n or f): The number of revolutions made per second. $n = \frac{1}{T}$.

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Additional Exercises

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