Unit 1 · 1.5 — Vectors and Motion in Two Dimensions

Everything to learn here

• Concept: In two-dimensional projectile motion, horizontal and vertical motions are completely independent of one another
• Concept: Time is the only scalar variable that links the horizontal and vertical kinematic equations in two-dimensional motion
• Concept: For ideal projectiles, horizontal velocity remains constant (ax = 0) while vertical acceleration is uniform downward (ay = -g)
• Concept: The vertical velocity of a projectile is exactly zero at the apex of its trajectory, but its vertical acceleration remains constant downward
• Concept: The path of a projectile without air resistance is a parabola
• Equation: $$v_{x0}=v_0\cdot \cos (\theta )\text{(Horizontal initial velocity component)}$$
• Equation: $$v_{y0}=v_0\cdot \sin (\theta )\text{(Vertical initial velocity component)}$$
• Equation: $$t=\frac{2\cdot v_0\cdot \sin (\theta )}{g}\text{(Time of flight for a projectile landing at its launch height)}$$
• Equation: $$h_{max}=\frac{v_0\cdot \sin (\theta {)}^2}{2\cdot g}\text{(Maximum height)}$$
• Equation: $$R=\frac{v_0^2\cdot \sin (2\cdot \theta )}{g}\text{(Horizontal range formula for a level surface)}$$
• Vocabulary: Projectile — An object launched into the air that is subject only to the downward force of gravity
• Vocabulary: Trajectory — The parabolic path followed by a projectile in two-dimensional motion
• Skill: Mathematical Routines — Decomposing an initial angled velocity vector into orthogonal x and y components to solve for the maximum height
• Skill: Experimental Design — Designing a lab to relate launch velocity to horizontal range or height, identifying independent and dependent variables
• Skill: Common Error Avoidance — Never plug a horizontal distance into a kinematic equation containing a vertical acceleration
• Concept: Any vector can be mathematically modeled as the resultant of two perpendicular components, and resolved into components using trigonometric functions relative to a chosen coordinate system
• Vocabulary: Vector Component — One of two perpendicular pieces of a vector, found using the sine and cosine of the angle it makes with an axis; the components' resultant reconstructs the original vector
• Concept: Projectile motion is a special case of two-dimensional motion in which acceleration is zero in one dimension and constant and nonzero in the other
• Skill: Analyze two-dimensional motion by resolving the object's path into independent horizontal and vertical kinematic components.
• Skill: Resolve vectors using sin(theta) = opposite/hypotenuse, cos(theta) = adjacent/hypotenuse, and a^2 + b^2 = c^2 to find components or magnitudes.
• Vocabulary: Vocabulary: Projectile Motion — A type of 2D motion with zero acceleration in one direction and constant acceleration in the perpendicular direction.

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