Lecture 4 1d Kinematics

Velocity-Time Graphs

• General equation: v = ds/dt

  • Velocity can vary with time (t)

  • Useful to plot v vs. t for 1-D motion analysis

  • Example: From position equation s = 1/2 t², we get v = t

Acceleration

• Definition: a = dv/dt

• Interpretation:

  • Acceleration (a) is the slope of the velocity/time curve

    • If a > 0, velocity (v) is increasing with time (t)

    • If a < 0, velocity (v) is decreasing with time

Notation and Terminology

• Mathematical Notation:

  • a = dv/dt = d/dt(ds/dt) = d²s/dt²

    • Here, ds/dt is the first derivative

  • Shorthand notation:

    • ṡ = ds/dt (velocity)

    • s̈ = d²s/dt² (acceleration)

    • v = ṡ

    • a = v̇ = s̈

Units of Acceleration

• Dimensions: Acceleration (a) has dimensions of (Length)/(Time)²

• SI Units: Measured in m/s² (meters per second squared)

• Example Calculation: For s(t) = At(T − t)

  • ṡ = A(T − 2t)

  • s̈ = -2A (constant acceleration)

Constant (Uniform) Acceleration Models

• Focus: One-dimensional motion with constant acceleration

• Important note: Formulas derived apply only to constant acceleration

Velocity/Time Diagram for Constant Acceleration

• Graph type: Linear graph with slope equal to acceleration (a)

• Motion Analysis: Starting from time t = 0 to later time t

• Initial parameters: Let initial velocity (u) be given

  • Slope relation: a = (v - u)/t

  • Final relation: v = u + at (for uniform acceleration)

Example of Average Acceleration

• Problem: Determine average acceleration of a car in m/s²

• Given conditions:

  • Initial velocity (u) = 0 km/h = 0 m/s

  • Final velocity (v) = 100 km/h = 100 x 10³ m / 3600 s

  • Time (t) = 3.7 s

• Calculation: a = (v - u)/t ≈ 7.5 m/s²

Differential Equations

• Definition: Any equation involving the derivative of an unknown function

• Application: Used to solve for velocity with uniform acceleration

  • Starting point: dv/dt = a

  • Subject to initial condition v(t = 0) = u

• Importance: Differential equations have broad applications across various fields (physics, engineering, biology, economics)

• Concept introduction: Integration is crucial to solving such equations.