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Basic Equations and Parabolic Path
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Projectile motion is a form of motion where an object moves in parabolic path; the path that the object follows is called its trajectory.
Learning Objective

Assess the effect of angle and velocity on the trajectory of the projectile; derive maximum height using displacement
Key Points
 Objects that are projected from, and land on the same horizontal surface will have a vertically symmetrical path.
 The time it takes from an object to be projected and land is called the time of flight. This depends on the initial velocity of the projectile and the angle of projection.
 When the projectile reaches a vertical velocity of zero, this is the maximum height of the projectile and then gravity will take over and accelerate the object downward.
 The horizontal displacement of the projectile is called the range of the projectile, and depends on the initial velocity of the object.
Terms

trajectory
The path of a body as it travels through space.

symmetrical
Exhibiting symmetry; having harmonious or proportionate arrangement of parts; having corresponding parts or relations.
Full Text
Projectile Motion
Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path. The path that the object follows is called its trajectory. Projectile motion only occurs when there is one force applied at the beginning on the trajectory, after which the only interference is from gravity. In a previous atom we discussed what the various components of an object in projectile motion are. In this atom we will discuss the basic equations that go along with them in the special case in which the projectile initial positions are null (i.e.
Initial Velocity
The initial velocity can be expressed as x components and y components:
In this equation,
Time of Flight
The time of flight of a projectile motion is the time from when the object is projected to the time it reaches the surface. As we discussed previously,
Acceleration
In projectile motion, there is no acceleration in the horizontal direction. The acceleration,
Velocity
The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time,
You can also use the Pythagorean Theorem to find velocity:
Displacement
At time, t, the displacement components are:
The equation for the magnitude of the displacement is
Parabolic Trajectory
We can use the displacement equations in the x and y direction to obtain an equation for the parabolic form of a projectile motion:
Maximum Height
The maximum height is reached when
where
Range
The range of the motion is fixed by the condition
Range of Trajectory
The range of a trajectory is shown in this figure.
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Key Term Reference
 Component
 Appears in these related concepts: Adding and Subtracting Vectors Using Components, Cathode Ray Tube, TV and Computer Monitors, and the Oscilloscope, and Position, Velocity, and Acceleration as a Function of Time
 acceleration
 Appears in these related concepts: Position, Displacement, Velocity, and Acceleration as Vectors, Scientific Applications of Quadratic Functions, and Centripetial Acceleration
 atom
 Appears in these related concepts: Overview of Atomic Structure, Description of the Hydrogen Atom, and Stable Isotopes
 displacement
 Appears in these related concepts: Calculus with Parametric Curves, Reference Frames and Displacement, and Introduction to Human Language
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 force
 Appears in these related concepts: Force of Muscle Contraction, Force, and First Condition
 gravity
 Appears in these related concepts: Motion with Constant Acceleration, Defining Graviational Potential Energy, and Properties of Electric Charges
 interference
 Appears in these related concepts: Holography, Thin Film Interference, and Superposition and Interference
 land
 Appears in these related concepts: Using Interference to Read CDs and DVDs, Key Points: Range, Symmetry, Maximum Height, and XRay Diffraction
 magnitude
 Appears in these related concepts: Multiplying Vectors by a Scalar, Roundoff Error, and Components of a Vector
 motion
 Appears in these related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 position
 Appears in these related concepts: Damped Harmonic Motion, Longitudinal Waves, and Graphical Interpretation
 velocity
 Appears in these related concepts: Velocity of Blood Flow, RootMeanSquare Speed, and Rolling Without Slipping
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Cite This Source
Source: Boundless. “Basic Equations and Parabolic Path.” Boundless Physics Boundless, 10 Oct. 2016. Retrieved 22 Feb. 2017 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/twodimensionalkinematics3/projectilemotion42/basicequationsandparabolicpath22610952/