Calculus Small Changes, Big Ideas

Section 6.1 Motion in the plane

• Projectile motion:

  🔗
  \begin{align*} x \amp = x_0 + v_x t\\ y \amp = y_0 + v_y t - \tfrac12 g t^2 \end{align*}
  🔗
• Circular motion:

  🔗
  \begin{align*} x \amp = h + r\cos t \\ y \amp = k + r\sin t \end{align*}
  🔗
• Velocity and acceleration in \(x\) and \(y\) directions

  🔗
  🔗
• We can combine these representations into a vector (which we’ll draw as an arrow):

  🔗
  • Position vector:

    \begin{equation*} \langle x(t), y(t) \rangle \end{equation*}

    🔗
    🔗
  • Velocity vector:

    \begin{equation*} \langle x'(t), y'(t) \rangle \end{equation*}

    🔗
    🔗
  • Acceleration vector:

    \begin{equation*} \langle x''(t), y''(t) \rangle \end{equation*}

    🔗
    🔗
  🔗
• Since we have multiple variables floating around, we can use ‘dot notation’ to denote derivatives with respect to time:

  🔗
  • Velocity vector:

    \begin{equation*} \langle \dot{x}, \dot{y} \rangle \end{equation*}

    🔗
    🔗
  • Acceleration vector:

    \begin{equation*} \langle \ddot{x}, \ddot{y} \rangle \end{equation*}

    🔗
    🔗
  🔗