Simultaneous Independent Vector Analysis Projectile Motion Simultaneous Independent Vector Analysis

Objectives Recognize that the vertical and horizontal motions of a projectile are simultaneous but independent. Learn and understand the terms associated with Projectile Motion Learn the difference between horizontal and angular projectile motion

What is Projectile Motion? T R A J E C T O R Y ( P A R A B O L A ) LOFT HEIGHT PROJECTILE RANGE

Gravity! When a projectile How Does it Work? is given an initial thrust, and we ignore air resistance, it moves through the air with only what acting on it? Gravity!

How Does Projectile Motion Work? We know that the horizontal motion of a launched ball does not affect its vertical motion. A projectile launched horizontally has no initial vertical velocity The vertical and horizontal motion of a projectile are independent of each other vi = 0 m/s

How Does Projectile Motion Work? Therefore a horizontally launched projectile has a vertical motion just like an object in freefall. That’s why objects launched horizontally will hit the ground at the same time as objects that are dropped from the same height. And horizontal motion just like an object in constant motion.

Angular Projectile Motion θ Y X

Solving 2-Dimensional Motion Problems Motion in two dimensions can be solved by breaking the problem into two interconnected one-dimensional problems. The two components are connected to the resultant by the Pythagorean Theorem. Δdx vx xi Δt xf 𝒗 𝒙 = ∆ 𝒅 𝒙 ∆𝒕 vx yi Δdy Δt vy yf vy 𝒗 𝒚 = ∆ 𝒅 𝒚 ∆𝒕 vR vx2 + vy2 = vR2

Solving Projectile Motion Problems Projectile motion can be divided into a vertical motion problem and a horizontal motion problem. 𝒗 𝒙 = ∆ 𝒅 𝒙 ∆𝒕 ∆ 𝒅 𝒙 The horizontal component can be solved like a constant motion problem ∆𝒕 𝒗 𝒙 ∆ 𝒅 𝒚 = 𝒗 𝒊 ∆𝒕+ 𝟏 𝟐 𝒂∆ 𝒕 𝟐 The vertical component can be solved like a freefall problem 𝒗 𝒊 𝒂 ∆𝒕 𝒗 𝒇 ∆ 𝒅 𝒚

Solving Projectile Motion Problems Both the vertical and horizontal components of a projectile are connected by time (t). The time for a projectile to hit the ground is the same for the vertical component and the horizontal component. tx tx = ty = tR tR ty

Concept Summary Projectile Motion involves an object moving horizontally while also in freefall. The trajectory of a projectile resembes a parabola The horizontal movement of a projectile resembles constant motion. The vertical movement of a projectile resembles freefall.

Concept Summary Time is the same for both vertical and horizontal components of projectile motion. If one object is launched horizontally at the same time as another is released to free fall from the same height, they will strike the ground at the same time. Angular projectile motion has an initial velocity that is not horizontal to the x-axis Horizontal projectile motion has an initial velocity that IS horizontal to the x-axis

vocabulary Projectile Motion – the motion of a moving body that is projected (launched). The horizontal motion and vertical motion of a projectile are independent of one another. The horizontal component of the velocity of the projectile remains constant while the vertical component is in free fall. The only variable that remains constant between the horizontal and vertical components are the time intervals for each, which are equal Horizontal Projectile Motion – projectile motion in which the projectile is launched at an angle that is parallel with the x-axis. Angular Projectile Motion – projectile motion in which the projectile is launched at an angle that is not parallel with the x-axis.

vocabulary Theta – the letter of the Greek alphabet (θ) that is used to represent the size of an angle Range – The total horizontal distance travelled by a projectile Trajectory – The path a projectile takes through space; resembles the shape of a parabola. Loft – The maximum height achieved by a projectile

vocabulary Simultaneous Equations – a set of two or more equations, each containing two or more variables whose values can simultaneously satisfy both or all of the equations in the set Example: 𝑣 𝑥 = ∆ 𝑑 𝑥 ∆𝑡 ∆ 𝑑 𝑦 = 𝑣 𝑖 ∆𝑡+ 1 2 𝑎∆ 𝑡 2 If the two equations above are simultaneous equations, then their common variable (∆𝑡) must be the same for both equations. Note: the number of unknown variables must equal to or less than the number of equations in order to solve.

vocabulary Substitution Method – A method of solving simultaneous equations with multiple unknowns: choose one of the variables common for both equations manipulate one of the equations to solve for that variable combine the two equations by substituting the manipulated equality for the same variable in the second equation. Solve the combined equation for the second unknown variable