Overview
This lecture explains how to use calculus, specifically derivatives and integrals, to calculate position, velocity, and acceleration in one-dimensional kinematics, and shows the calculus origins of common kinematic equations.
Derivatives in Kinematics
- A derivative measures the rate of change or the slope of a function.
- The derivative of position with respect to time (dx/dt or x') gives velocity.
- The derivative of velocity with respect to time (dv/dt or v') gives acceleration.
- Calculus is used because physical quantities often change with time, not always remaining constant.
- The power rule for derivatives: d/dt(xⁿ) = n·xⁿ⁻¹.
Integrals in Kinematics
- An integral (or antiderivative) can be used to determine original functions from their derivatives.
- The integral of acceleration with respect to time gives velocity.
- The integral of velocity with respect to time gives position.
- Power rule for integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1).
- Integrals add a constant of integration, representing initial conditions.
Derivation of Kinematic Equations (CamiCues)
- With constant acceleration (a), integrating gives velocity: v = v₀ + at.
- Integrating velocity gives position: x = x₀ + v₀t + ½at².
- These commonly used kinematic equations originate directly from calculus.
- The third kinematic equation’s origin will be discussed in a future lesson.
Key Terms & Definitions
- Derivative — Rate of change of a function; gives slope.
- Integral — Operation to find the original function from its rate of change; area under the curve.
- Power Rule — Shortcut for taking derivatives and integrals of power functions.
- Constant of Integration — Constant added during integration, representing initial values.
Action Items / Next Steps
- Review the power rule for both derivatives and integrals.
- Ensure understanding of how the kinematic equations are derived from calculus.
- Prepare questions for clarification if needed.