Tuesday, February 2, 2010

Projectile Motion

In calculus, a model for projectile motion with no friction is considered, and a "parabolic trajectory" is obtained. If the initial velocity is [Graphics:Images/ProjectileMotionMod_gr_1.gif] and [Graphics:Images/ProjectileMotionMod_gr_2.gif] is the initial angle to the horizontal, then the parametric equations for the horizontal and vertical components of the position vector are

(1) [Graphics:Images/ProjectileMotionMod_gr_3.gif],
and
(2) [Graphics:Images/ProjectileMotionMod_gr_4.gif].

Solve equation (1) for t and get [Graphics:Images/ProjectileMotionMod_gr_5.gif], then replace this value of t in equation (2) and the result is

[Graphics:Images/ProjectileMotionMod_gr_6.gif],

which is an equation of a parabola.

The time [Graphics:Images/ProjectileMotionMod_gr_7.gif] required to reach the maximum height is found by solving [Graphics:Images/ProjectileMotionMod_gr_8.gif]:

[Graphics:Images/ProjectileMotionMod_gr_9.gif],
yields
[Graphics:Images/ProjectileMotionMod_gr_10.gif],

and the maximum height is

[Graphics:Images/ProjectileMotionMod_gr_11.gif].

The time [Graphics:Images/ProjectileMotionMod_gr_12.gif] till impact is found by solving [Graphics:Images/ProjectileMotionMod_gr_13.gif], which yields [Graphics:Images/ProjectileMotionMod_gr_14.gif], and for this model, [Graphics:Images/ProjectileMotionMod_gr_15.gif]. The range is found by calculating [Graphics:Images/ProjectileMotionMod_gr_16.gif]:

[Graphics:Images/ProjectileMotionMod_gr_17.gif].

For a fixed initial velocity [Graphics:Images/ProjectileMotionMod_gr_18.gif], the range [Graphics:Images/ProjectileMotionMod_gr_19.gif] is a function of [Graphics:Images/ProjectileMotionMod_gr_20.gif], and is maximum when [Graphics:Images/ProjectileMotionMod_gr_21.gif].

Numerical solution of second order D. E.'s

This module illustrates numerical solutions of a second order differential equation. First, we consider the special case where the projectile is fired vertically along the y-axis and has no horizontal motion, i. e. x(t)= 0. The effect of changing the amount of air drag or air resistance is investigated. It is known that the drag force acting on an object which moves very slowly through a viscous fluid is directly proportional to the velocity of that object. However, there are examples, such as Millikan's oil drop experiment, when the drag force is proportional to the square of the velocity. Further investigations into the situation could involve the Reynolds number.

Math-Models (Projectile Motion I):

The following mathematical models are are considered.
(i). No air resistance [Graphics:Images/ProjectileMotionMod_gr_22.gif], and
[Graphics:Images/ProjectileMotionMod_gr_23.gif].

(ii). Air resistance proportional to velocity [Graphics:Images/ProjectileMotionMod_gr_24.gif], and
[Graphics:Images/ProjectileMotionMod_gr_25.gif].

(iii). Air resistance proportional to the square of the velocity [Graphics:Images/ProjectileMotionMod_gr_26.gif], for the ascent, and [Graphics:Images/ProjectileMotionMod_gr_27.gif], for the descent, and
[Graphics:Images/ProjectileMotionMod_gr_28.gif][Graphics:Images/ProjectileMotionMod_gr_29.gif]

(iv). Air resistance proportional to the [Graphics:Images/ProjectileMotionMod_gr_30.gif] power of the velocity
[Graphics:Images/ProjectileMotionMod_gr_31.gif][Graphics:Images/ProjectileMotionMod_gr_32.gif]

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Introductory Methods of Numerical Analysis