Lecture5 1d Kinematics

1. Integration Fundamentals

• Integration Definition

  • Opposite process to differentiation.

  • If ( f(t) ) is a function of ( t ), then ( g(t) = \frac{df}{dt} ) defines the derivative function.

  • For any constant ( C ):

    • [ g(t) = \frac{df(t)}{dt} = \frac{d}{dt}(f(t) + C) ]

• Indefinite Integral

  • Defined as:

    • [ \int g(t) dt = f(t) + C ]

  • Thus,

    • [ \int \frac{df}{dt} dt = f(t) + C ]

2. Notation and Examples

• Integral Notation

  • Symbol ( \int ) is a historical elongated ( S ) indicating a sum.

• Example 1: Integration of 1

  • Find ( \int 1 dt ):

    • Here, ( g(t) = 1 ) gives ( f(t) = t ).

    • Result:

      • [ \int 1 dt = t + C ]

• Example 2: Integration of t

  • Find ( \int t dt ):

    • Here, ( g(t) = t ) and ( f(t) = \frac{1}{2} t^2 ).

    • Result:

      • [ \int t dt = \frac{1}{2} t^2 + C ]

3. General Case of Integration of ( t^n )

• Example 3: Generalizing Integration

  • Find ( \int t^n dt ) for all ( n ):

    • Using ( g(t) = t^n ) leads to ( f(t) = \frac{1}{n+1} t^{n+1} ) (for ( n
      eq -1 )).

    • Result:

      • [ \int t^n dt = \frac{1}{n + 1} t^{n + 1} + C ]

    • For ( n = -1 ):

      • [ \int t^{-1} dt = \int \frac{1}{t} dt ]

4. Applications in Kinematics

• Differential Equation for Velocity

  • Equation:

    • [ \frac{dv}{dt} = a ]

  • Integrating both sides gives

    • [ \int dv = \int a dt ]

  • Result:

    • [ v = at + C ]

    • Fixed using initial velocity ( v(t=0) = u ):

      • Leads to [ v = u + at ]

• Position Formula

  • Starting from:

    • ( ds/dt = u + at )

  • Integrate:

    • [ s = \int (u + at) dt = ut + \frac{1}{2} a t^2 + C ]

    • Assuming ( s(0) = 0 ) gives ( C = 0 )

    • Thus:

      • [ s = ut + \frac{1}{2} a t^2 ]

5. Relationship Among Variables in Uniform Acceleration

• Third Relationship Derivation

  • From ( v = u + at ):

    • Rearranging:

      • [ t = \frac{v - u}{a} ]

    • Substituting into ( s ):

      • Leads to [ s = \frac{u(v - u)}{a} + \frac{1}{2} a \left(\frac{v - u}{a}\right)^2 ]

      • Resulting in:

        • [ s = \frac{1}{2} a (v^2 - u^2) ]

        • Thus:

          • [ v^2 = u^2 + 2as ]

6. Summary of Key Formulas for Kinematics

• Core Formulas for 1-Dimensional Uniform Acceleration:

  • [ s = ut + \frac{1}{2} at^2 ]

  • [ v = u + at ]

  • [ v^2 = u^2 + 2as ]

• Note:

  • These formulas are applicable only to 1-dimensional motion under uniform (constant) acceleration.