Lecture Notes: Solving a Physics Problem & Introduction to Projectile Motion

Solving the Steel Ball Problem

• Problem Description:

  • Steel ball A slides on a horizontal rod towards an electromagnet.
  • Force of attraction follows inverse square law.
  • Acceleration is given by (k / (L - x)^2).
  • Initial conditions: Released from rest at (x_0), initial velocity = 0.
  • Goal: Determine the velocity when it strikes the pole face.

• Solution Approach:

  • Use the formula (a dx = v dv).
  • Integration bounds:
    • Lower bound: (x_0)
    • Upper bound: (L - D/2) (consider center of the ball)
  • Integration leads to:
    • ( \frac{1}{2} V_s^2 = \text{integral result} )
  • Solving the integration gives:
    • ( V^2 = \frac{2}{D} - \frac{1}{L} )
  • Additional considerations for constant (k).

Introduction to Planar Curvilinear Motion

• Concepts:
  • Coordinates system used: rectangular, polar, normal, and tangential.
  • Example: car moving on a curve (e.g., on a ramp).
  • Velocity and acceleration components:
    • Velocity: (\vec{v} = \frac{d\vec{R}}{dt})
    • Speed: (\frac{ds}{dt})
    • Acceleration: (\vec{a} = \frac{d\vec{v}}{dt})

Rectangular Coordinate System

• Position & Velocity:
  • Position vector: ( \vec{R}(t) = x(t)\hat{i} + y(t)\hat{j} )
  • Velocity: ( \vec{v}(t) = \dot{x}\hat{i} + \dot{y}\hat{j} )
  • Acceleration: ( \vec{a} = \ddot{x}\hat{i} + \ddot{y}\hat{j} )

Projectile Motion

• Basic Assumptions:

  • No air resistance
  • No rotation of the Earth affecting the projectile
  • Only force acting is gravity

• Equations of Motion:

  • Horizontal velocity (v_x = v_0 \cos(\theta))
  • Vertical velocity (v_y = v_0 \sin(\theta) - gt)
  • Horizontal position: (x = v_0 \cos(\theta)t + x_0)
  • Vertical position: (y = -\frac{1}{2} g t^2 + v_0 \sin(\theta) t + y_0)

• Time-independent Formula:

  • ( y = \tan(\theta) x - \frac{g}{2 v_0^2 \cos^2(\theta)} x^2 + y_0 )

• Key Parameters:

  • Maximum height (when vertical velocity = 0)
  • Time of flight
  • Range of the projectile

• Example: Calculation of range and maximum height for a projectile launched at an angle of 60 degrees.

Practical Applications

• Use in sports: Basketball shots, baseball throws
• Military applications: Trajectory calculations for projectiles

Upcoming Topics

• More problem-solving in the next class
• Use of mathematical tools for solving complex equations
• Transition to normal and tangential coordinate system understanding