CHAPTER 3: MOTION IN TWO OR THREE DIMENSIONS

Introduction to 2D Motion Analysis
When an object moves in two dimensions, its position, velocity, and acceleration are described by vectors with $x$- and $y$-components that vary with time.

Calculating Average and Instantaneous Velocity (Example 3.1)

• Position: Described by $x(t)$ and $y(t)$. Distance from origin is the magnitude of the position vector $\Vert\vec{r}\Vert = \sqrt{x^2 + y^2}$.

• Displacement Vector ($\Delta \vec{r}$): $\vec{r}_f - \vec{r}_i$.

• Average Velocity Vector ($\vec{v}_{\text{avg}}$): $\frac{\Delta \vec{r}}{\Delta t}$.

• Instantaneous Velocity Vector ($\vec{v}(t)$): The derivative of the position vector, with components $v_x(t) = \frac{dx}{dt}$ and $v_y(t) = \frac{dy}{dt}$.

Calculating Average and Instantaneous Acceleration (Example 3.2)

• Average Acceleration Vector ($\vec{a}_{\text{avg}}$): $\frac{\Delta \vec{v}}{\Delta t}$.

• Instantaneous Acceleration Vector ($\vec{a}(t)$): The derivative of the velocity vector, with components $a_x(t) = \frac{dv_x}{dt}$ and $a_y(t) = \frac{dv_y}{dt}$.

Calculating Parallel and Perpendicular Components of Acceleration (Example 3.3)

• Parallel Component ($a_{\parallel}$): Acts along the direction of velocity, changing speed. Calculated as $a_{\parallel} = \frac{\vec{a} \cdot \vec{v}}{\Vert\vec{v}\Vert}$.

• Perpendicular Component ($a_{\perp}$): Acts perpendicular to velocity, changing direction. Calculated as $a_{\perp} = \sqrt{\Vert\vec{a}\Vert^2 - a_{\parallel}^2}$.

Projectile Motion (Section 3.3)

• Definition: Motion of an object launched and moving freely under gravity, assuming negligible air resistance. Its path is typically a parabola.

• Key Principles:

  • Moves in a single vertical plane.

  • Constant downward acceleration ($a_x = 0$, $a_y = -g \approx -9.8 \text{ m/s}^2$).

  • Horizontal and vertical motions are independent: horizontal velocity is constant, vertical velocity changes due to gravity.

• Equations of Motion (launch from origin at angle $\alpha_0$ with speed $v_0$):

  • Initial Velocity Components: $v_{0x} = v_0 \cos \alpha_0$, $v_{0y} = v_0 \sin \alpha_0$.

  • Velocity at time $t$: $v_x(t) = v_{0x}$, $v_y(t) = v_{0y} - gt$.

  • Position at time $t$: $x(t) = v_{0x} t$, $y(t) = v_{0y} t - \frac{1}{2}gt^2$.

• Top of Trajectory: vertical velocity $v_y = 0$, but vertical acceleration $a_y = -g$ still applies.

Key Examples and Concepts:

• Object Projected Horizontally (Ex 3.6): Demonstrates calculating position and velocity for a projectile launched horizontally from a height.

• Height and Range of a Projectile I (Ex 3.7): Shows methods to find position, velocity, maximum height, and total horizontal range for an angled launch.

• Height and Range of a Projectile II (Ex 3.8):

  • Maximum Height Formula: $h = \frac{v_0^2 \sin^2 \alpha_0}{2g}$.

  • Horizontal Range Formula: $R = \frac{v_0^2 \sin(2\alpha_0)}{g}$.

  • Conditions for Maximums (for a given $v_0$):

  • Maximum Height at $\mathbf{90^\circ}$.

  • Maximum Horizontal Range at $\mathbf{45^\circ}$ (if launch and landing are at the same height).

• Different Initial and Final Heights (Ex 3.9): Involves solving a quadratic equation for time when a projectile lands at a different height than its launch point.

• The Zookeeper and the Monkey (Ex 3.10): Illustrates the independence of horizontal and vertical motion; an object aimed directly at a target that simultaneously falls under gravity will always hit the target.