Kinematics in Two Dimensions

Chapter 3: Kinematics in Two Dimensions; Vectors

3-1 Vectors and Scalars

• Definition of a Vector:

  • A vector is a mathematical object that has both magnitude (size) and direction.

  • Representation: Vectors are typically represented by arrows. The length of the arrow is proportional to the magnitude, and the arrow points in the vector's direction.

  • Equality of Vectors: Two vectors are considered equal if they have the same magnitude and the same direction, regardless of where their starting points are located in space.

  • Example vector quantities include:

  • Displacement: $\Delta \vec{r} = \vec{r}<em>f - \vec{r}</em>i$, denoting change in position.

  • Velocity: $\vec{v} = \frac{d\vec{r}}{dt}$, the rate of change of position.

  • Force: $\vec{F} = m\vec{a}$, an influence that changes motion.

  • Acceleration: $\vec{a}$, the rate of change of velocity.

• Definition of a Scalar:

  • A scalar is a quantity that has only magnitude and no direction.

  • Algebra: Scalars follow ordinary laws of arithmetic.

  • Example scalar quantities include:

  • Mass ($m$): Quantity of matter.

  • Time ($t$): Duration of events.

  • Temperature ($T$): Measure of thermal energy.

  • Distance and Speed: The magnitude-only versions of displacement and velocity.

3-2 Addition of Vectors—Graphical Methods

• Properties of Vector Addition:

  • Commutative Law: The order of addition does not matter: $\vec{A} + \vec{B} = \vec{B} + \vec{A}$.

  • Associative Law: $\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}$.

• Two-Dimensional Vector Addition:

  • For perpendicular vectors (e.g., North and East), the magnitude of the resultant vector ($R$) is found using the Pythagorean Theorem:
    $R = \sqrt{A^2 + B^2}$

  • The angle ($\theta$) relative to the x-axis is found using:
    $\theta = \tan^{-1}\left(\frac{A<em>y}{A</em>x}\right)$

• Graphical Representation:

  • Tail-to-Tip Method: Place the tail of the second vector at the tip of the first. The resultant connects the tail of the first to the tip of the last.

  • Parallelogram Method: Place both tails at the same origin. The resultant is the diagonal of the parallelogram formed by these two vectors.

3-3 Subtraction of Vectors, and Multiplication of a Vector by a Scalar

• Subtraction of Vectors:

  • Defined as adding the additive inverse. If $\vec{A}$ is a vector, $-\vec{A}$ has the same magnitude but points in the opposite direction ($180^\circ$ difference).

  • Equation: $\vec{D} = \vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

• Multiplication by a Scalar:

  • Multiplying a vector $\vec{V}$ by a scalar $c$ yields $c\vec{V}$.

  • If c > 0, the direction is unchanged, but the magnitude is scaled by $c$.

  • If c < 0, the direction is reversed, and the magnitude is scaled by $|c|$.

3-4 Adding Vectors by Components

• Definition of Components:

  • Any vector $\vec{V}$ can be resolved into specific parts along the axes of a coordinate system, typically $V<em>x$ and $V</em>y$.

  • Using Unit Vectors: $\vec{V} = V<em>x\hat{i} + V</em>y\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors of magnitude 1 along the x and y axes.

• Trigonometric Resolution:

  • For a vector at angle $\theta$ with the positive x-axis:
    $V<em>x = V \cos(\theta)$ $V</em>y = V \sin(\theta)$

• Algebraic Addition:

  • If $\vec{R} = \vec{A} + \vec{B}$, then:
    $R<em>x = A</em>x + B<em>x$ $R</em>y = A<em>y + B</em>y$

  • The final magnitude and direction are:
    $R = \sqrt{R<em>x^2 + R</em>y^2}$
    $\theta = \text{atan2}(R<em>y, R</em>x)$

3-5 Projectile Motion

• Assumptions of Projectile Motion:

  • Air resistance is neglected.

  • Acceleration due to gravity ($g = 9.80\text{ m/s}^2$) is constant and directed downward.

  • The horizontal ($x$) and vertical ($y$) motions are independent of each other.

• Components of Motion:

  • Horizontal: Constant velocity ($a<em>x = 0$). $v</em>x = v_{x0}$.

  • Vertical: Constant acceleration ($a_y = -g$). It behaves like free-fall.

3-6 Solving Projectile Motion Problems

• Kinematic Equations for Projectiles:

  • Horizontal ($x$):
    $x = v<em>{x0}t$ $v</em>x = v<em>{x0} = v</em>0 \cos(\theta_0)$

  • Vertical ($y$):
    $y = v<em>{y0}t - \frac{1}{2}gt^2$ $v</em>y = v<em>{y0} - gt$ $v</em>y^2 = v<em>{y0}^2 - 2g(y - y</em>0)$
    $v<em>{y0} = v</em>0 \sin(\theta_0)$

• Strategic Points:

  • At the maximum height, the vertical velocity component is zero: $v_y = 0$.

  • The Time of Flight for a symmetric trajectory (landing at the same height) is $t = \frac{2v<em>0 \sin(\theta</em>0)}{g}$.

3-7 Projectile Motion Is Parabolic

• By substituting $t = \frac{x}{v<em>{x0}}$ into the vertical displacement equation, we derive the path equation: $y = (\tan \theta</em>0)x - \left( \frac{g}{2v<em>0^2 \cos^2 \theta</em>0} \right)x^2$

• This takes the standard form of a parabola: $y = Ax - Bx^2$.

3-8 Relative Velocity

• Subscript Notation:

  • The first subscript refers to the object, the second to the reference frame.

  • Example: $\vec{v}_{PA}$ is the velocity of Person (P) relative to Air (A).

• Vector Addition of Velocities:

  • $\vec{v}<em>{PS} = \vec{v}</em>{PA} + \vec{v}_{AS}$

  • (Velocity of Person relative to Shore = Velocity of Person relative to Air + Velocity of Air relative to Shore).

• The Inverse Rule: $\vec{v}<em>{AB} = -\vec{v}</em>{BA}$.