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Basic Equations and Parabolic Path
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 Objectives

Assess the effect of angle and velocity on the trajectory of the projectile

Explain how to 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
Basic Equations and Parabolic Path
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, u stands for initial velocity magnitude and
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, T depends on the initial velocity magnitude and the angle of the projectile:
In projectile motion, there is no acceleration in the horizontal direction.
The acceleration, a, in the vertical direction is just due to gravity, also known as free fall:
Velocity
The horizontal velocity remains constant, but the vertical velocity varies linearly, because the acceleration is constant. At any time, t, the velocity is:
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
The equation of a parabola is
Maximum Height
The maximum height is reached when
where t_{h} stands for the time it takes to reach maximum height. From the displacement equation we can find the maximum height
Range
The range of the motion is fixed by the condition
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Key Term Reference
 Component
 Appears in this related concepts: Adding and Subtracting Vectors Using Components, Position, Velocity, and Acceleration as a Function of Time, and Cathode Ray Tube, TV and Computer Monitors, and the Oscilloscope
 acceleration
 Appears in this related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and Applications and ProblemSolving
 atom
 Appears in this related concepts: Early Ideas about Atoms, Stable Isotopes, and John Dalton and Atomic Theory
 displacement
 Appears in this related concepts: Calculus with Parametric Curves, Reference Frames and Displacement, and Interference
 equation
 Appears in this related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 force
 Appears in this related concepts: Work Done by a Variable Force, Driven Oscillations and Resonance, and Glancing Collisions
 gravity
 Appears in this related concepts: Motion with Constant Acceleration, Properties of Electric Charges, and Defining Graviational Potential Energy
 interference
 Appears in this related concepts: Interference and Diffraction, Holography, and Superposition and Interference
 land
 Appears in this related concepts: XRay Diffraction, Using Interference to Read CDs and DVDs, and Key Points: Range, Symmetry, Maximum Height
 magnitude
 Appears in this related concepts: Roundoff Error, Multiplying Vectors by a Scalar, and Components of a Vector
 motion
 Appears in this related concepts: Motion Diagrams, TwoComponent Forces, and Moving Source
 position
 Appears in this related concepts: Damped Harmonic Motion, Longitudinal Waves, and Graphical Interpretation
 velocity
 Appears in this related concepts: RootMeanSquare Speed, Arc Length and Speed, and Tangent and Velocity Problems
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Source: Boundless. “Basic Equations and Parabolic Path.” Boundless Physics. Boundless, 12 Dec. 2014. Retrieved 23 Apr. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/twodimensionalkinematics3/projectilemotion42/basicequationsandparabolicpath22610952/