Kinematics Notes

Addition of Vectors

• Head to Tail Method:
  • Graphical method to find the resultant vector of two or more vectors.
  • Tail of the second vector connects to the head of the first vector.
  • The resultant vector (R) is drawn from the tail of the first to the head of the last vector.
  • $R = A + B$
• Commutative Law:
  • The order of vector addition doesn't matter: $A + B = B + A$
• Associative Law:
  • The sum of vectors remains the same regardless of grouping.

Multiplication and Division of Vectors

• Vectors can be multiplied/divided by scalars.
  • Positive scalar: changes magnitude.
  • Negative scalar: changes magnitude and direction.

Vector Components and Addition

• Rectangular Components Method:
  • Resolve vectors into x and y components.
  • $V<em>{1x} = V</em>1 \cos \theta_1$
  • $V<em>{1y} = V</em>1 \sin \theta_1$
  • $V<em>{2x} = V</em>2 \cos \theta_2$
  • $V<em>{2y} = V</em>2 \sin \theta_2$
  • Resultant vector components:
    • $R<em>x = V</em>{1x} + V<em>{2x} = V</em>1 \cos \theta<em>1 + V</em>2 \cos \theta_2$
    • $R<em>y = V</em>{1y} + V<em>{2y} = V</em>1 \sin \theta<em>1 + V</em>2 \sin \theta_2$
  • Magnitude of resultant vector: $R = \sqrt{R<em>x^2 + R</em>y^2}$
  • Direction of resultant vector: $\theta = \tan^{-1} \frac{R<em>y}{R</em>x}$

Scalar Product (Dot Product)

• $A \cdot B = AB \cos \theta$
• Commutative: $A \cdot B = B \cdot A$
• Perpendicular vectors: $A \cdot B = 0$
• Unit vectors: $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$
• Parallel vectors: $A \cdot B = AB$
• Unit vectors: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$
• Antiparallel vectors: $A \cdot B = -AB$
• Self-product: $A \cdot A = A^2$
• Rectangular components: $A \cdot B = A<em>x B</em>x + A<em>y B</em>y + A<em>z B</em>z$

Vector Product (Cross Product)

• $A \times B = AB \sin \theta \hat{n}$
• Non-commutative: $A \times B = -B \times A$
• Associative: $(mA) \times B = A \times (mB) = m(A \times B)$
• Distributive: $A \times (B + C) = A \times B + A \times C$
• Perpendicular vectors: $|A \times B| = AB$
• Parallel vectors: $A \times B = 0$
• Unit vectors: $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
• $\hat{i} \times \hat{j} = \hat{k}, \quad \hat{j} \times \hat{k} = \hat{i}, \quad \hat{k} \times \hat{i} = \hat{j}$
• Rectangular components: $A \times B = (A<em>y B</em>z - A<em>z B</em>y) \hat{i} + (A<em>z B</em>x - A<em>x B</em>z) \hat{j} + (A<em>x B</em>y - A<em>y B</em>x) \hat{k}$

Speed and Velocity

• Speed: distance/time, scalar quantity.
  • Average, uniform, and instantaneous speeds exist.
• Velocity: displacement/time, vector quantity.
  • Average velocity: change in position/time interval.
  • Instantaneous velocity: $V = \lim_{ \Delta t \to 0} \frac{\Delta s}{\Delta t}$

Acceleration

• Change in velocity with respect to time.
• $Acceleration = \frac{Change \, in \, velocity}{time \, taken}$
• Positive, negative (retardation), or zero.
  • Uniform acceleration: constant magnitude and direction.
  • Non-uniform acceleration: changing magnitude or direction.
  • Average acceleration: $a_{avg} = \frac{\Delta v}{\Delta t}$
  • Instantaneous acceleration: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Equations of Motion (Uniform Acceleration)

• $V<em>f = V</em>i + at$
• $s = V_i t + \frac{1}{2} a t^2$
• $V<em>f^2 = V</em>i^2 + 2as$

Projectile Motion

• Motion in a plane under gravity.
• Assumptions: constant gravity, negligible air resistance.
• Horizontal motion: constant velocity.
  • $a_x = 0$
  • $V<em>x = V</em>{0x} = V_0 \cos \theta$
  • $x = x<em>0 + V</em>{0x} t$
• Vertical motion: constant acceleration.
  • $a_y = -g$
  • $V<em>y = V</em>{0y} - gt = V_0 \sin \theta - gt$
  • $y = y<em>0 + V</em>{0y}t - \frac{1}{2}gt^2$
• Time to reach max height: $t = \frac{V_0 \sin \theta}{g}$
• Total time of flight: $T = \frac{2V_0 \sin \theta}{g}$
• Maximum height: $H = \frac{V_0^2 \sin^2 \theta}{2g}$
• Range: $R = \frac{V_0^2 \sin 2\theta}{g}$
• Maximum range at 45 degrees.