- #1
mysticjbyrd
- 18
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Homework Statement
This is a real world problem and not a problem in a book, so if you feel I am missing some information just let me know.
A ball is flying at 25 m/s
The wind is blowing perpendicular to the trajectory at .44704 m/s
[tex]\rho = \frac{1.17 kg}{{m}^3} [/tex]
[tex]Cd = 1/2 [/tex]
[tex]Diameter_{ball} = 0.04593 m [/tex]
[tex]Mass_{ball} = 0.04267 kg [/tex]
[tex]gravity = 9.8 \frac{m}{s}[/tex]
X-Y-Z plane
X-axis is straight towards the target
Y-axis is up in the air
Z-axis is left or right to the target
Using time steps of .1 seconds,
What is the force the wind applies on the ball?
How far will the ball fly off the trajectory in 2 seconds?
Homework Equations
[tex]F_{d} = .5*rho*A*{v}^2[/tex]
[tex]F_{d} = drag force[/tex]
[tex]\rho = air density [/tex]
[tex]v = velocity[/tex]
A = frontal area_{ball}
I am using,
[tex]A=\pi/4*{d}^2[/tex]
The Attempt at a Solution
Force of wind on the ball,
Using time steps of .1 s, Find all velocities to find total velocity.
[tex]v_{x}=25\frac{m}{s}[/tex]
[tex]
v_{y} = a*t => v_{y}=9.8*.1 = 0.98\frac{m}{s}[/tex]
Force applied on ball from wind
[tex] F_{ball}=F_{wind}-F_{drag}[/tex]
[tex]
F_{w} = .5*\rho*Cd*{V_{w}^2*A => F_{w}= .5*1.17*.5*{.44704}^2*\frac{\pi}{4}*{0.04593}^2 => F_{w}=0.0000968504 N
[/tex]
[tex]
F_{w}=m_{ball}*a_{ball} => \frac{F}{m}=a => \frac{0.0000968504}{0.04267} = 0.00226975\frac{m}{s^2} => v=at => {0.00226975}*.1 = 0.000226975\frac{m}{s}[/tex]
Velocity of ball no drag
[tex]v_{bnd} = 0.000226975\frac{m}{s}[/tex]
[tex]F_{d}= .5*\rho*Cd*{V_{b}^2*A => F_{d}= .5*1.17*.5*{0.000226975}^2*\frac{\pi}{4}*{0.04593}^2 => F_{d}= {{2.49669}^{-11}}N
[/tex]
[tex]
F_{ball}=F_{wind}-F_{drag} = 0.0000968504-{2.49669^{-11}}= {0.0000968504} N [/tex]
[tex]
F_{b}=m_{ball}*a_{ball} => \frac{F}{m}=a => \frac{{0.0000968504}}{0.04267} = 0.00226975\frac{m}{s^2} => v=at => {{0.00226975}}*.1 = {0.000226975}\frac{m}{s}[/tex]
[tex]v_{z}= 0.000226975\frac{m}{s}[/tex]
Force of Drag on ball X-axis
[tex]F_{d}= .5*1.17*.5*{25}^2*\frac{\pi}{4}*{0.04593}^2 = 0.302893N => a=\frac{F}{m} => \frac{0.302893}{0.04267} = 7.09849\frac{m}{s^2} => v=at => 7.09849 * .1 = .70949\frac{m}{s}
[/tex]
Total Velocity
[tex]
v_{xy} ={({v_{ball_x}^2+v_{ball_y}^2)}^\frac{1}{2} => v_{xy}=({{25}^2+{.98}^2)}^\frac{1}{2} = 25.0192\frac{m}{s}[/tex]
[tex]
V_{total}= {({v_{ball_xy}^2+v_{ball_z}^2)}^\frac{1}{2} => v_{total}=({{25.0192}^2+({8.9776*{10}^{-6})^2)}^\frac{1}{2} = 25.0192\frac{m}{s}[/tex]
If you would prefer to just use variables that would be fine with me... Since I am new to this style of formatting, it took me a long time to post this, and I do not expect someone else to do the same. However, it would be amazing if someone did.
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