Analytical Mechanics Projectiles

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    Analytical Mechanics

    Projectiles:Are They Coming or

    Going?Original Article Author: James S. Walker

    By:

    M.H. Malek Khotaee

    P. Babaee

    M. Jamshidi

    M.R. Zangeneh

    In The Name of God

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    In t roduct i

    onIn this presentation, we discuss about radialdistance of a projectile.

    Radial distance is the size of Radial (Location)

    Vector of a projectile which means the distance

    between projectile and throw origin at any time ofits flight.

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    (,)

    = +

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    =

    2 cos

    = /

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    = + = (

    2)/

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    This behavior begins at a critical angel at:

    ~And continues for all projectiles launched with

    angels between and 90.

    90

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    A plot of ras a function of time (t) shows the same

    behavior.

    = =

    1

    2

    = +

    = +

    1

    4

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    The Crit ical

    Angel =

    (

    )

    For small angels, the solutions are not real

    meaning that the projectile steadily moves away

    form the origin cause:

    = (

    ) = .

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    The Retu rn

    El l ipse

    +

    =

    0

    -(

    ) (

    )

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    The Return Ellipse

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    Between these two

    points, the projectile

    is moving closer to

    the origin:

    <

    The Return Ellipse

    The Route of Projectiles Motion

    =

    =

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    The Single Point:

    =

    The Return Ellipse

    The Route of Projectiles Motion

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    Conclus ion1- Ask one person to shoot a basketball while a

    second person videotapes the ball from a vantage

    point looking over the first persons shoulder. The

    diameter of the ball can be measured frame by

    frame, and then plotted. Alternatively, videotape

    the balls flight at right angels to its plane of

    motion. In this case, the distance from the launch

    site to the ball, can be measured directly on each

    frame.

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    2- Finally, it may be possible to detect radial

    oscillation using the Doppler effect. Throw a ballthat produces a tone and listen for the frequency to

    fall, rise, and fall again. The effect may be too

    small to detect with the human ear, but this and theprevious suggestions could make for interesting

    undergraduate experiments.