Lecture 2 1d Kinematics

Velocity and the Derivative

1. Motion along a Line

  • Observing position of an object at:

    • Initial time: t

    • Slightly later time: t + ∆t

  • Define position functions:

    • s(t): position at time t

    • s(t + ∆t): position at time t + ∆t

  • ∆t refers to a small increment (infinitesimal) in time, not multiplication.

2. Change in Position

  • Over the interval ∆t, the change in position is:

    • ∆s = s(t + ∆t) − s(t)

  • Average velocity can be calculated as:

    • Average velocity = ∆s / ∆t

    • This represents the slope of the secant line on the graph of s versus t.

3. Instantaneous Velocity

  • Taking the limit as ∆t → 0 gives:

    • v = lim (∆t→0) ∆s / ∆t

  • Geometrically, this represents the slope of the tangent line to the curve s(t).

  • Definition of Derivative:

    • v = ds/dt = lim (∆t→0) ∆s/∆t

  • Instantaneous velocity describes the motion of the particle at any moment in time.

4. Examples of Velocity Calculation

4.1 Constant Velocity Example

  • Given:

    • s(t) = Ct (where C is a constant)

  • Compute ∆s:

    • ∆s = s(t + ∆t) − s(t) = C(t + ∆t) − Ct = C∆t

  • Average velocity:

    • ∆s / ∆t = C

  • Thus:

    • v = lim (∆t→0) ∆s/∆t = C

4.2 Quadratic Position Example

  • Given:

    • s(t) = At² (where A is a constant)

  • Compute ∆s:

    • ∆s = A(t + ∆t)² − At² = 2At∆t + A(∆t)²

  • Instantaneous velocity:

    • v = lim (∆t→0) (2At + A∆t) = 2At

4.3 Increasing Velocity Example

  • Use A = 1 m.s⁻² as a constant:

  • This implies that velocity is increasing as time increases.

4.4 Nonlinear Position Example

  • Given:

    • s(t) = A(t)(T − t) (where A and T are constants)

  • Compute ∆s:

    • ∆s = AT(t + ∆t) − A(t + ∆t)² − At(T − t)

    • Expands to:

    • ∆s = AT∆t − 2At∆t − A(∆t)²

  • Instantaneous velocity:

    • v = lim (∆t→0) (AT − 2At − A∆t) = AT − 2At

Summary

  • Understanding motion involves calculating changes in position over time, which leads to definitions of average and instantaneous velocity using derivatives. The derivative indicates how position changes concerning time and illustrates the particle's motion characteristics.