Vectors and Motion in Two Dimensions

Using Vectors

• A vector is a quantity that has both magnitude (size) and direction.
• Vectors can represent a particle's velocity, where the particle's speed is the magnitude and the direction is indicated by an arrow.
• The displacement vector represents the straight-line connection from the initial to the final position, regardless of the actual path taken.
• Two vectors are equal if they have the same magnitude and direction, irrespective of their starting points.

Vector Addition

• Net displacement $\vec{d}$ is the result of multiple displacements (e.g., $\vec{d}<em>1$ followed by $\vec{d}</em>2$).
• The net displacement $\vec{d}$ is the sum of individual displacements: $\vec{d} = \vec{d}<em>1 + \vec{d}</em>2$
• The sum of two vectors is called the resultant vector.
• Vector addition is commutative: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$
• Tip-to-tail and parallelogram rules are methods for visualizing vector addition.

Multiplication by a Scalar

• Multiplying a vector by a positive scalar changes its magnitude but not its direction.
• Multiplying a vector by zero results in the zero vector (a vector with zero length).
• A vector cannot have a negative magnitude.
• Multiplying a vector by a negative number reverses its direction.
• Multiplying a vector by -1 reverses its direction without changing its magnitude.

Subtracting Vectors

• To subtract vector $\vec{B}$ from $\vec{A}$:
  1. Draw $\vec{A}$.
  2. Place the tail of $\vec{B}$ at the tip of $\vec{A}$.
  3. Draw an arrow from the tail of $\vec{A}$ to the tip of $\vec{B}$. This represents $\vec{A} - \vec{B}$.

Coordinate Systems and Vector Components

• A coordinate system is an artificially imposed grid for making quantitative measurements.
• Cartesian coordinates are commonly used, featuring perpendicular axes with positive and negative ends separated by zero at the origin.
• Component vectors are vectors parallel to the coordinate axes that, when summed, equal the original vector.
• For a vector $\vec{A}$ in an xy-coordinate system, the component vectors are $\vec{A}<em>x$ and $\vec{A}</em>y$, where $\vec{A} = \vec{A}<em>x + \vec{A}</em>y$.

Determining Vector Components

• The absolute value of the x-component (Ax) is the magnitude of the component vector $\vec{A}_x$.
• The sign of Ax is positive if $\vec{A}_x$ points in the positive x-direction and negative if it points in the negative x-direction.
• The y-component (Ay) is determined similarly.

Working with Components

• Vectors can be added using their components.
• If $\vec{C} = \vec{A} + \vec{B}$, then the components of $\vec{C}$ are the sums of the components of $\vec{A}$ and $\vec{B}$.
• $C<em>x = A</em>x + B_x$
• $C<em>y = A</em>y + B_y$

Tilted Axes

• For motion on a slope, it is convenient to align the x-axis along the slope, creating a tilted coordinate system.
• Finding components with tilted axes is similar to using standard axes.

Accelerated Motion on a Ramp

• When a crate slides down a frictionless ramp tilted at an angle $\theta$, its acceleration is parallel to the surface.
• Choosing the x-axis along the ramp simplifies the analysis.
• The acceleration parallel to the ramp is a component of the free-fall acceleration: $a = g \sin(\theta)$.

Motion in Two Dimensions

• In two dimensions, an object moves in a plane, and its displacement, velocity, and acceleration vectors can all change.

Vectors on Motion Diagrams

• In two dimensions, an object's displacement is a vector.
• The velocity vector is the displacement vector multiplied by a scalar, pointing in the direction of the displacement.

Acceleration in Two Dimensions

• The vector definition of acceleration is an extension of the one-dimensional version: $\vec{a} = \frac{\Delta \vec{v}}{\Delta t}$.
• Acceleration occurs whenever there is a change in velocity, which can be a change in magnitude (speed) or direction.

Projectile Motion

• Projectile motion is two-dimensional motion under the influence of gravity alone (neglecting air resistance).
• The vertical and horizontal components of projectile motion are independent of each other.
• The vertical component of acceleration is the familiar free-fall acceleration (g), while the horizontal component is zero.

Analyzing Projectile Motion

• The launch angle is the angle of the initial velocity above the horizontal.
• Projectile motion consists of uniform motion at constant velocity in the horizontal direction and free-fall motion in the vertical direction.

Projectile Motion Equations

• The kinematic equations for projectile motion combine constant-acceleration motion vertically and constant-velocity horizontally.
• Horizontal:
  • $x<em>f = x</em>i + (v<em>x)</em>i \Delta t$
  • $(v<em>x)</em>f = (v<em>x)</em>i = \text{constant}$
• Vertical:
  • $y<em>f = y</em>i + (v<em>y)</em>i \Delta t - \frac{1}{2} g (\Delta t)^2$
  • $(v<em>y)</em>f = (v<em>y)</em>i - g \Delta t$
• $\Delta t$ is the same for both horizontal and vertical components.

Problem-Solving Approach: Projectile Motion

1. Strategize: Treat horizontal and vertical motions separately.
2. Prepare:
   • Make simplifying assumptions (e.g., neglecting air resistance).
   • Draw a visual overview, including a pictorial representation.
   • Establish a coordinate system with the x-axis horizontal and the y-axis vertical.
   • Horizontal acceleration will be zero, and vertical acceleration will be free fall (-g).
   • Draw the initial velocity vector and find its x- and y-components.
   • Define symbols and list known values; identify what the problem is trying to find.
3. Solve: Use kinematic equations for horizontal and vertical components.
4. Assess: Check units, reasonableness, and ensure the question is answered.

Circular Motion

• Uniform circular motion involves moving at a constant speed in a continuously changing direction.
• Objects in uniform circular motion are not at constant velocity because their velocity vectors change direction.

Uniform Circular Motion

• For circular motion at a constant speed, the acceleration vector (a) points toward the center of the circle.
• Acceleration directed towards the center of a circle is called centripetal acceleration.

UCM Equations

• $a = \frac{v^2}{r}$

Relative Motion

• The velocity of an object depends on the observer's frame of reference.
• $\vec{v}_{RA}$ represents the velocity of the Runner relative to Amy.
• $\vec{v}_{AC}$ represents the velocity of Amy relative to Carlos.
• $\vec{v}<em>{RC} = \vec{v}</em>{RA} + \vec{v}_{AC}$ represents the velocity of the Runner relative to Carlos.