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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

Explain how to derive maximum height using displacement

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

symmetrical
Exhibiting symmetry; having harmonious or proportionate arrangement of parts; having corresponding parts or relations.

trajectory
The path of a body as it travels through space.
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
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, Wavelength, Freqency in Relation to Speed, and Rotational Collisions
 acceleration
 Appears in these related concepts: Centripetial Acceleration, Position, Displacement, Velocity, and Acceleration as Vectors, and The Second Law: Force and Acceleration
 atom
 Appears in these related concepts: The Law of Multiple Proportions, Stable Isotopes, and John Dalton and Atomic Theory
 displacement
 Appears in these related concepts: Reference Frames and Displacement, Interference, and Introduction to Human Language
 equation
 Appears in these related concepts: A General Approach, Equations and Inequalities, and Equations and Their Solutions
 force
 Appears in these related concepts: Work, Force, and Force of Muscle Contraction
 gravity
 Appears in these related concepts: Properties of Electric Charges, Defining Graviational Potential Energy, and Key Points: Range, Symmetry, Maximum Height
 interference
 Appears in these related concepts: Holography, Diffraction, and The Michelson Interferometer
 land
 Appears in these related concepts: XRay Diffraction, Relative Velocity, and Using Interference to Read CDs and DVDs
 magnitude
 Appears in these related concepts: Scalars vs. Vectors, Multiplying Vectors by a Scalar, and Components of a Vector
 motion
 Appears in these related concepts: Newton and His Laws, Mechanical Work and Electrical Energy, and Friction: Static
 position
 Appears in these related concepts: Motion with Constant Acceleration, Motion Diagrams, and Graphical Interpretation
 velocity
 Appears in these related concepts: Rolling Without Slipping, RootMeanSquare Speed, and Applications and ProblemSolving
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Source: Boundless. “Basic Equations and Parabolic Path.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 07 Oct. 2015 from https://www.boundless.com/physics/textbooks/boundlessphysicstextbook/twodimensionalkinematics3/projectilemotion42/basicequationsandparabolicpath22610952/