Motion in Two and Three Dimensions: Kinematics and Projectile Dynamics

Position and Velocity Vectors

• Definition of Position Vector: The position vector of an object, denoted as $\mathbf{r}$, is defined as the vector that originates from the origin of the coordinate system and extends to the point $P$ where the object is currently located.
• Components of the Position Vector: The Cartesian coordinates $(x, y, z)$ of point $P$ represent the specific components of the position vector along the $x$, $y$, and $z$ axes. This can be expressed as:
  • $\mathbf{r} = x\hat{i} + y\hat{j} + z\hat{k}$
• Average Velocity Vector ($\mathbf{v}_{avg}$): This vector describes the displacement of an object as it moves from an initial position (at time $t_i$) to a final position (at time $t_f$).
  • Time Interval ($\Delta t$): The duration of the motion is defined as $\Delta t = t_f - t_i$.
  • Calculation: Average velocity is the ratio of the displacement vector $\Delta \mathbf{r}$ to the time interval $\Delta t$:
    • $\mathbf{v}_{avg} = \frac{\Delta \mathbf{r}}{\Delta t}$
  • Directional Property: The displacement vector $\Delta \mathbf{r}$ always points in the exact same direction as the average velocity vector $\mathbf{v}_{avg}$.
• Instantaneous Velocity Vector ($\mathbf{v}$): This is defined as the limit of the average velocity as the time interval $\Delta t$ approaches zero.
  • Calculus Definition: The velocity vector is equal to the derivative of the position vector with respect to time:
    • $\mathbf{v} = \frac{d\mathbf{r}}{dt}$

The Acceleration Vector

• Average Acceleration Vector ($\mathbf{a}_{avg}$): Defined for an object whose velocity changes by $\Delta \mathbf{v}$ during a specific time interval $\Delta t$.
  • Calculation: It is the ratio of the change in velocity to the change in time:
    • $\mathbf{a}_{avg} = \frac{\Delta \mathbf{v}}{\Delta t}$
• Directional Property: The average acceleration vector $\mathbf{a}_{avg}$ points in the same direction as the velocity change vector $\Delta \mathbf{v}$.
• Instantaneous Acceleration Vector ($\mathbf{a}$): This is the limit of the average acceleration as $\Delta t$ approaches zero.
  • Calculus Definition: The instantaneous acceleration is the derivative of the velocity vector with respect to time:
    • $\mathbf{a} = \frac{d\mathbf{v}}{dt}$

Motion in a Circular Path

• Fundamental Principle of Acceleration: A particle undergoes acceleration whenever its velocity vector changes over time. This change can occur in magnitude, direction, or both.
• Tangential Component of Acceleration ($\mathbf{a}_t$): This component results specifically from a change in the magnitude (speed) of the velocity vector.
• Centripetal (Radial) Component of Acceleration ($\mathbf{a}_c$ or $\mathbf{a}_r$): This component results specifically from a change in the direction of the velocity vector.
  • Direction: The centripetal component is always directed toward the center of the circular path.
• General Acceleration in Non-Uniform Circular Motion: For a particle in circular motion where speed is not constant, the total acceleration vector is the sum of the tangential and radial acceleration vectors:
  • $\mathbf{a} = \mathbf{a}_t + \mathbf{a}_r$
• Uniform Circular Motion: This is a specific case of two-dimensional motion where a particle moves along a circular path at a constant speed ($v$).
• Period of Motion ($T$): The period is defined as the total time required for the particle to complete exactly one full revolution around the circular path.
• Formula for Period in Uniform Circular Motion:
  • $T = \frac{2\pi r}{v}$
  • $r$ = the radius of the circular path.
  • $v$ = the constant speed of the particle.

Two-Dimensional Motion with Constant Acceleration

• Vector Description: The motion of a particle subject to constant acceleration is described by kinematic equations. These equations can be separated into independent horizontal and vertical components.
• Component Independence: By writing vectors in terms of their $x$ and $y$ components, two distinct sets of kinematic equations are obtained: one for the $x$-axis and one for the $y$-axis.
• Connection Parameter: Time ($t$) is the critical parameter that connects the motion along the $x$-axis with the motion along the $y$-axis.
• Horizontal Motion (x-motion) Equations:
  1. $v_{xf} = v_{xi} + a_x t$
  2. $x_f = x_i + v_{xi} t + \frac{1}{2} a_x t^2$
  3. $v_{xf}^2 = v_{xi}^2 + 2 a_x (x_f - x_i)$
• Vertical Motion (y-motion) Equations:
  1. $v_{yf} = v_{yi} + a_y t$
  2. $y_f = y_i + v_{yi} t + \frac{1}{2} a_y t^2$
  3. $v_{yf}^2 = v_{yi}^2 + 2 a_y (y_f - y_i)$

Projectile Motion

• Acceleration Conditions: In projectile motion, the object is subject only to the free-fall acceleration due to gravity.
  • $a_x = 0$: There is zero acceleration in the horizontal direction, resulting in constant velocity motion.
  • $a_y = g$: The vertical direction is governed by free-fall motion (where $g$ is the acceleration due to gravity).
• Kinematic Equations for Projectile Motion:
  • Horizontal ($x$-axis) motion:
    1. $v_{xf} = v_{xi}$
    2. $x_f = x_i + v_{xi} t$
  • Vertical ($y$-axis) motion:
    1. $v_{yf} = v_{yi} + a_y t$
    2. $y_f = y_i + v_{yi} t + \frac{1}{2} a_y t^2$

Relative Velocity

• General Velocity Vector Rules:
  • Notation: Velocities are written with double subscripts to denote the reference frame. For example, $\mathbf{v}_{AB}$ means the "velocity of body $A$ relative to body $B$."
  • Addition Rule: When adding relative velocities, the first letter of any subscript in the sum must be the same as the last letter of the preceding subscript:
    • $\mathbf{v}_{AC} = \mathbf{v}_{AB} + \mathbf{v}_{BC}$
  • Sum Identification: The first letter of the first velocity's subscript and the second letter of the last velocity's subscript determine the identity of the resultant relative velocity.
  • Negative Relationship: The velocity of body $A$ relative to body $B$ is the exact negative of the velocity of body $B$ relative to body $A$:
    • $\mathbf{v}_{AB} = -\mathbf{v}_{BA}$