Study Notes on Relative Velocity and Constraint Motion

Fundamentals of Relative Velocity

• Relative Velocity Definition:

  • The velocity of a particle $A$ with respect to a particle $B$ is defined as the velocity of $A$ as observed from the reference frame of $B$.
  • Vector Expression: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B = \vec{v}_A + (-\vec{v}_B)$.

• General Vector Subtraction and Resultants:

  • If the angle between two vectors $\vec{A}$ and $\vec{B}$ is $\theta$, the magnitude of the resultant vector $R = A + B$ is typically given by $\sqrt{A^2 + B^2 + 2AB \cos(\theta)}$.
  • When calculating relative velocity magnitudes, the subtraction logic applies: $v_{AB} = \sqrt{v_A^2 + v_B^2 - 2v_A v_B \cos(\theta)}$.

The Rain-Umbrella Problem

• Problem Setup:

  • A person is moving with velocity $\vec{v}_m$ and rain is falling with velocity $\vec{v}_r$.
  • The person must hold an umbrella at a specific angle to avoid getting wet, which corresponds to the direction of the velocity of rain relative to the man ($\vec{v}_{rm}$).

• Governing Equation:

  • $\vec{v}_{rm} = \vec{v}_r - \vec{v}_m = \vec{v}_r + (-\vec{v}_m)$.

• Calculating the Angle:

  • If the rain falls vertically and the man moves horizontally, the angle $\theta$ with the vertical at which the umbrella should be tilted is found using trigonometry:     $\tan(\theta) = \frac{v_m}{v_r}$$\theta = \tan^{-1}(\frac{v_m}{v_r})$

• Relative Magnitude:

  • The speed of rain as perceived by the man is:     $v_{rm} = \sqrt{v_r^2 + v_m^2}$

River Boat Problem

• Variables and Notation:

  • $S$: Width of the river.
  • $u$: Velocity of the river flow (drift velocity).
  • $v$: Velocity of the boat in still water (or relative to the water).
  • $\theta$: The angle the boat's velocity vector makes with the normal (vertical) to the river bank.

• Time Taken to Cross the River:

  • Crossing depends only on the velocity component perpendicular to the bank:     $t = \frac{S}{v \cos(\theta)}$

• Drift of the Boat ($d$):

  • Drift is the horizontal displacement along the bank caused by the river flow and the boat's horizontal component:     $d = (u - v \sin(\theta))t$$d = (u - v \sin(\theta)) \frac{S}{v \cos(\theta)}$

Relative Velocity of Multiple Particles and Shortest Distance

• Relative Velocity between Two Particles:

  • For two particles 1 and 2 moving in a plane, the velocity of 1 relative to 2 is: $\vec{v}_{12} = \vec{v}_1 - \vec{v}_2 = \vec{v}_1 + (-\vec{v}_2)$.

• Minimum Distance ($d_{min}$):

  • To find the minimum distance between two moving particles, the problem is analyzed in the relative frame (typically making particle 2 stationary).
  • Particle 1 moves along a straight line in the relative frame. The shortest distance is the perpendicular length from the stationary particle 2 to the relative velocity vector line of particle 1.

Connected and Constraint Motion

• Principles of Constraint:

  • This topic deals with systems connected by ropes, chains, or rods.
  • Key assumptions: The connector is inextensible, massless, and remains stretched.

• Velocity Constraint Principle:

  • The velocity of each particle along the length of the rope, rod, or chain must be the same to prevent the connector from stretching or breaking.
  • If two points $A$ and $B$ are on a rigid rod, and $v_A$ and $v_B$ are their respective velocities making angles with the rod's length, then:     $v_A \cos(\theta_1) = v_B \cos(\theta_2)$

Numerical Examples in Kinematics

• Example 1: Rod Moving Against a Wall:

  • A rod is leaned against vertical and horizontal axes. One end moves with velocity $v$ at an angle of $53^\circ$ to the vertical. The other end moves at $5\,m/s$ at an angle of $37^\circ$ to the horizontal.
  • Using the constraint equation:     $v \cos(53^\circ) = 5 \cos(37^\circ)$$v \times \frac{3}{5} = 5 \times \frac{4}{5}$$v = \frac{20}{3}\,m/s \approx 6.67\,m/s$
  • Note: In specific handwritten annotations (Page 7-8), alternate results like $v = 25\,m/s$ or $v = 20\,m/s$ are noted based on different input values (e.g., $15\,m/s$ instead of $5\,m/s$).

• Example 2: Simple Pulley System:

  • Calculation for a moving pulley where $v_p = \frac{v_1 + v_2}{2}$.
  • Given: Pulley moves up at $5\,m/s$ ($v_p = +5$), one block moves up at $2\,m/s$ ($v_1 = +2$).
  • Calculation:     $5 = \frac{2 + v_2}{2}$$10 = 2 + v_2$$v_2 = 8\,m/s$

• Example 3: Complex Pulley and Block Configuration:

  • Analysis of multiple blocks and pulleys where the total length of the string is constant ($l = x_1 + x_2 + ...$).
  • For a block and pulley system (Page 11):     $v = 10\,m/s$ (given as a specific instance).     Formula for length derivative: $\sum v_i \cos(\theta_i) = 0$

• Example 4: Relative Displacement Calculation (Page 9):

  • Finding a final velocity or displacement where $2v = 1.6\,m/s$ suggests $v = 0.8\,m/s$ (labeled as $1.6$ in some contexts).