Forest UCM PnCP ProjMotion

Projectile Motion

Friction depends linearly on velocity

Projectile motion describes the path a mass moving in two dimensions. An example of which is the motion of a projectile shot out of a cannon with an initial velocity [math]v_0[/math] with an angle of inclination [math]\theta[/math].

When the motion in each dimension is independent, the kinematics are separable giving you two equations of motion that depend on the same time.

Using our solutions for the horizontal and vertical motion when friction depends linearly on velocity (Forest_UCM_PnCP_LinAirRes) we can write :

[math]x= \frac{m}{b} v_i \left ( 1-e^{-\frac{b}{m}t} \right )[/math]

in the y-direction however, the directions are changed to represent an object moving upwards instead of falling

Newton's second law for falling

[math]\sum \vec{F}_{ext} = mg -bv = m \frac{dv}{dt}[/math]

becomes

[math]\sum \vec{F}_{ext} = -mg +bv = m \frac{dv}{dt}[/math]

for a rising projectile

This changes the signs in front of the [math]v_t[/math] terms such that

[math]y= v_t t + \frac{1}{b}\left ( v_0 - v_t) \right ) \left ( 1- e^{-bt} \right ) [/math]

becomes

[math]y = -v_t t + \frac{1}{b}\left ( v_0 + v_t) \right ) \left ( 1- e^{-bt} \right ) [/math]

We now have a system governed by the following system of two equations

[math]y = \frac{1}{b}\left ( v_0 + v_t) \right ) \left ( 1- e^{-bt} \right ) -v_t t[/math]

Forest_UCM_PnCP#Projecile_Motion