Engineering › Projectiles ›

Frictionless

The flight of a Frictionless Projectiles

Controller: lodmore

Contents

Interface
Height
Max Height
Impact Velocity
Position
Page Comments

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Interface

C++

Excel

Overview

With a Frictionless system the influence of air on the projectile is ignored. On earth this leads to a optimistic estimate for the distance travelled by the projective, although on an airless planet/moon these calculations would be appropriate.

In this module, the initial velocity of the projectile is R, released at an angle $\inline \theta$ at a height h. i.e.

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The only force acting on a frictionless projectile is that of gravity. As can be seen from the diagram the initial launch velocity,R, is resolved into it's vertical and horizontal components. As there are no forces in the horizontal direction, the horizontal component of the initial velocity lasts for the duration of the flight. Therefore the main equations of motion need only be applied in the vertical direction:

$s=ut -{ \frac{1}{2}}{g t^2}$

(2)

$v^2=u^2 + 2as$

(3)

$v=u+at$

(4)

Where

u is the initial vertical velocity [m/s]
v is the final vertical velocity [m/s]
s is the vertical distance traveled [m]
t is the time taken [s]
a is acceleration [m/s²]

Authors

James Bateman (2007)

Height

doubleheight(	double	`R`
	double	`theta`
	double	`t`
	double	`h` `= 0`	)

In the vertical direction the only force acting on the projectile is Gravity, g, which is assumed to be constant. With an initial vertical velocity of $\inline u= R \sin \theta$ , then Eq (1) gives

$s(t)=R \sin\theta \cdot t -{ \frac{1}{2}}{g t^2}$

(5)

which is the vertical height of the projectile at any given time.

Parameters

R	is the initial velocity of the projectile [m/s]
theta	is the angle the projectile is released from the horizontal [degrees]
t	the time from when the projectile is launched
h	Default value = 0

Source Code

This module is private, for owner's use only.

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Max Height

doublemax_height(	double	`R`
	double	`theta`	)

At its greatest height, the vertical velocity is zero, therefore from Eq (2) the maximum vertical height is:

$s=\frac{R^2\sin^2\theta}{2g}$

(6)

We can show from Eq(3) that the time to reach this maximum is

$t=\frac{R\sin\theta}{g}$

(7)

Parameters

R	is the initial velocity of the projectile [m/s]
theta	is the angle the projectile is released from the horizontal [degrees]

Source Code

This module is private, for owner's use only.

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Impact Velocity

doubleimpact_velocity(	double	`R`
	double	`theta`
	double	`h`	)

The velocity of the projectile as it hits the ground is

$V=\sqrt{v_{horz}^2 + v_{vertl}^2}$

(8)

where

v_horz = $\inline R \cos \theta$
v_vert is computed from Eq (2), with the initial assumptions that u = 0 (at maximum height) and height of the projectile from its final landing point is

$s=h+\frac{R^2 sin^2 \theta}{2 g}$

(9)

Therefore

$v_{vert}^2 = 2g(\frac{h+R^2\sin^2\theta}{2g})$

(10)

$v_{vert}=\sqrt{h+R^2\sin^2\theta}$

(11)

Parameters

R	is the initial velocity of the projectile [m/s]
theta	is the angle the projectile is released from the horizontal [degrees]
h	the initial vertical height of the projectile

Source Code

This module is private, for owner's use only.

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Position

doubleposition(	double	`R`
	double	`theta`
	double	`x`
	double	`h` `= 0`	)

To conveniently plot the position of the projectile, it is necessary to compute the height of the projectile, y, for a given horizontal position x.

Therefore at a given x position, the time taken to reach that point is merely:

$t=\frac{x}{R\cos\theta}$

(12)

From Eq(3) and maximum flight time of the projectile is:

$t_{max}=\frac{1}{g}\left [\sqrt{h+R^2\sin^2\theta} + R \sin\theta \right ]$

(13)

which serves as an upper limit for Eq(11)

Using the height function (above), y is computed.

Parameters

R	is the initial velocity of the projectile [m/s]
theta	is the angle the projectile is released from the horizontal [degrees]
x	the horizontal position of the projectile.
h	the initial vertical height of the projectile.

Source Code

This module is private, for owner's use only.

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Real	cc_projectilesHeight (Real R, Real theta, Real t, Real h)
Real	cc_projectilesMax_height (Real R, Real theta)
Real	cc_projectilesImpact_velocity (Real R, Real theta, Real h)
Real	cc_projectilesPosition (Real R, Real theta, Real x, Real h)

double	`height` (double R, double theta, double t, double h=0) The vertical height of a projectile at any given time
double	`max_height` (double R, double theta) The maximum height achieved by a projectile
double	`impact_velocity` (double R, double theta, double h) The impact velocity of the projectile as it hits the ground
double	`position` (double R, double theta, double x, double h=0) The projectile height at a given horizontal position x