2D Mathematical Methods, Vector Mechanics, Projectile Motion, and Wave Equations

Scalar and Vector Quantities

• Scalar Quantity:

  • Definition: A physical quantity that possesses magnitude only and has no direction.

  • Examples: Distance, speed, mass, temperature, work, power, energy, pressure.

• Vector Quantity:

  • Definition: A physical quantity that possesses both magnitude and direction.

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

• Special Case - Electric Current:

  • Electric current has both magnitude and direction.

  • However, it is classified as a scalar quantity because it does not obey the laws of vector addition.

Types of Vectors

• Equal Vectors:

  • Definition: Vectors that have the exact same magnitude and the exact same direction.

• Null Vector or Zero Vector:

  • Definition: A vector having zero magnitude.

  • Example: The displacement vector of a body when it completes a round trip and returns to its initial position.

• Unit Vector:

  • Definition: A vector having unit magnitude in a specified or particular direction.

  • Mathematical expression: \n\mathbf{\hat{A}} = \frac{\mathbf{A}}{|\mathbf{A}|}\n

Laws of Vector Addition and Derivations

• Triangle Law of Vector Addition:

  • Statement: If two vectors are represented in magnitude and direction by the two sides of a triangle taken in the same order, then their resultant vector is represented in magnitude and direction by the third side of the triangle taken in reverse order ($\mathbf{R} = \mathbf{A} + \mathbf{B}$).

• Parallelogram Law of Vector Addition:

  • Statement: If two vectors are represented in magnitude and direction by two adjacent sides of a parallelogram, then their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram passing through their common point of intersection ($\mathbf{R} = \mathbf{A} + \mathbf{B}$).

• Derivation of Resultant Vector Magnitude ($R$):

  • Consider two vectors $\mathbf{A}$ and $\mathbf{B}$ inclined at an angle $\theta$ to each other.

  • From geometry of the triangle formed by extending vector $\mathbf{A}$:

    • $OS^2 = ON^2 + SN^2$

    • $OS^2 = (OP + PN)^2 + SN^2$

  • Substituting $OP = A$, $PN = B\cos(\theta)$, and $SN = B\sin(\theta)$:

    • $R^2 = (A + B\cos(\theta))^2 + (B\sin(\theta))^2$

    • $R^2 = A^2 + 2AB\cos(\theta) + B^2\cos^2(\theta) + B^2\sin^2(\theta)$

    • $R^2 = A^2 + 2AB\cos(\theta) + B^2(\cos^2(\theta) + \sin^2(\theta))$

    • $R^2 = A^2 + B^2 + 2AB\cos(\theta)$

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

• Direction of Resultant Vector:

  • Let $\alpha$ be the angle made by the resultant $\mathbf{R}$ with vector $\mathbf{A}$:

    • $\tan(\alpha) = \frac{SN}{ON}$

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

Projectile Motion

• Path and Velocity Components:

  • The trajectory or path followed by a projectile is a parabola.

  • Horizontal component of initial velocity: $u\cos(\theta)$

  • Vertical component of initial velocity: $u\sin(\theta)$

• Proof that the Path of a Projectile is a Parabola:

  • Horizontal distance covered in time $t$:

    • $x = u\cos(\theta)t \implies t = \frac{x}{u\cos(\theta)}$

  • Vertical displacement in time $t$:

    • $y = u\sin(\theta)t - \frac{1}{2}gt^2$

  • Substituting the expression for $t$ into the vertical displacement equation:

    • $y = u\sin(\theta)\left(\frac{x}{u\cos(\theta)}\right) - \frac{1}{2}g\left(\frac{x}{u\cos(\theta)}\right)^2$

    • $y = \tan(\theta)x - \left(\frac{g}{2u^2\cos^2(\theta)}\right)x^2$

  • This equation is of the standard form $y = ax + bx^2$, which defines a parabola.

• Time of Flight ($T$):

  • Using kinematic equation $S = ut + \frac{1}{2}at^2$ for total vertical motion ($S = 0$):

    • $0 = u\sin(\theta)T - \frac{1}{2}gT^2$

    • $\frac{1}{2}gT^2 = u\sin(\theta)T$

    • $T = \frac{2u\sin(\theta)}{g}$

• Horizontal Range ($R$):

  • Calculated as the product of horizontal velocity and total time of flight:

    • $R = \text{Horizontal velocity} \times \text{Time of flight}$

    • $R = u\cos(\theta) \times \frac{2u\sin(\theta)}{g}$

    • $R = \frac{u^2(2\sin(\theta)\cos(\theta))}{g}$

    • $R = \frac{u^2\sin(2\theta)}{g}$

• Maximum Height ($H$):

  • Using kinematic relation $v^2 - u^2 = 2as$ for vertical displacement at the peak where vertical velocity is zero:

    • $0 - u^2\sin^2(\theta) = -2gH$

    • $2gH = u^2\sin^2(\theta)$

    • $H = \frac{u^2\sin^2(\theta)}{2g}$

• Key Properties of Projectile Motion:

  • The horizontal range $R$ is maximum when the angle of projection \theta = 45^\circ$.\n * The horizontal range R$is identical for complimentary angles of projection$\theta$and$(90^\circ - \theta).\n * At the highest point of the trajectory, the vertical component of velocity becomes zero (u\sin(\theta) = 0).\n * Acceleration acts solely in the vertical direction: a_y = -g$.

Relation Between Linear Velocity and Angular Velocity

• Derivation:

  • By definition of angle in radians:

    • $\text{Angle} = \frac{\text{arc}}{\text{radius}}$

    • $\Delta\theta = \frac{\Delta r}{r}$

    • $\Delta r = r\Delta\theta$

  • Dividing both sides by time interval $\Delta t$:

    • $\frac{\Delta r}{\Delta t} = r\frac{\Delta\theta}{\Delta t}$

  • Taking the limit as $\Delta t \rightarrow 0$:

    • $\frac{dr}{dt} = r\frac{d\theta}{dt}$

  • Since linear speed $v = \frac{dr}{dt}$ and angular velocity $\omega = \frac{d\theta}{dt}$:

    • $v = r\omega$

Centripetal Acceleration

• Definition:

  • The acceleration experienced by a body moving in a circular path directed towards the centre of the circle along the radius is called centripetal acceleration.

• Derivation:

  • Using similar triangle geometric relations for velocity vectors and position vectors:

    • $\frac{\Delta v}{v} = \frac{\Delta r}{r}$

    • $\Delta v = v\frac{\Delta r}{r}$

  • Dividing by $\Delta t$:

    • $\frac{\Delta v}{\Delta t} = \frac{v}{r}\frac{\Delta r}{\Delta t}$

  • Taking the limit as $\Delta t \rightarrow 0$:

    • $\frac{dv}{dt} = \frac{v}{r}\frac{dr}{dt}$

  • Substituting acceleration $a = \frac{dv}{dt}$ and velocity $v = \frac{dr}{dt}$:

    • $a = \frac{v}{r} \times v$

    • $a = \frac{v^2}{r}$

Wave Mechanics and Wave Speed Equations

• Speed of Longitudinal Waves:

  • Speed of a longitudinal wave in a solid bar:

    • $v = \sqrt{\frac{Y}{\rho}}$

    • Where $Y$ is Young's modulus and $\rho$ is density.

  • General speed of a longitudinal wave in a medium:

    • $v = \sqrt{\frac{B}{\rho}}$

    • Where $B$ is Bulk modulus and $\rho$ is density.

  • Newton's formula for the speed of a longitudinal wave in an ideal gas:

    • $v = \sqrt{\frac{P}{\rho}}$

    • Where $P$ is pressure and $\rho$ is density.

• Superposition and Standing Waves:

  • Superposition of two waves travelling in opposite directions:

    • $y_1 = a\sin(kx - \omega t)$

    • $y_2 = a\sin(kx + \omega t)$

    • $y = y_1 + y_2 = a\sin(kx - \omega t) + a\sin(kx + \omega t)$

    • $y = (2a\sin(kx))\cos(\omega t)$

  • Amplitude of the resulting standing wave:

    • $A = 2a\sin(kx)$

  • Condition for Nodes (points of zero amplitude):

    • $\text{Amplitude} = 0$

    • $2a\sin(kx) = 0 \implies \sin(kx) = 0$

    • $kx = n\pi$

    • Substituting $k = \frac{2\pi}{\lambda}$:

    • $\frac{2\pi}{\lambda}x = n\pi \implies x = \frac{n\lambda}{2}$

  • Condition for Antinodes (points of maximum amplitude):

    • $\text{Amplitude} = \text{maximum}$

    • $2a\sin(kx) = \pm 2a$

    • $\sin(kx) = \pm 1$