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Deriving displacement, velocity and acceleration projecticle motion 
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#1
Aug2009, 05:10 PM

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1. An object of mass, M, is projected into the air with an initial vertical component of velocity, Vo. If the air resistance is proportional to the instantaneous velocity, with the constant of proportionality being, K, derive the equations for the displacement, velocity and acceleration as functions of time.
appologies for the way this is wrote if anyone has a better way of notating this it would be much appreciated. my work m dv/dt=(mgkv) so first I got my V's on one side and my T's on the other mdv/(mg+kv)=dt then added the integral sign to this equation and integrated it too get m/k ln(mg+kv)=t+c next I tidied it up mg+kv=e^((k/m)*t)+c that goop in the middle is meant to be exp to the power of whats in the brackets mg+kv=ce^((k/m)*t) movin the c for more tidiness now I tried to get V on its own V=ce^((k/m)*t)mg/k sub in t=0 and V=Vo Vo=Cmg/k therfore c is C=Vo + mg/k subbing back into my previous I got v=(Vo + mg/k)e^((k/m)*t)mg/k that is my velocity part right thats pretty much it for me any help would be appreciated for displacement and acceleration[/b] 


#2
Aug2009, 06:03 PM

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#3
Aug2009, 06:55 PM

P: 9

Sorry I just took that part from my notes I just have first
mdv/(mg+kv)=dt then adding the intergration symbol integrate mdv/(mg+kv)= integrate dt So I am quessing it's with respect to time as its to be as a function of time I really have only a vague idea whats going on here. I believe its a separable equation yes no 


#4
Aug2009, 07:03 PM

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P: 40,652

Deriving displacement, velocity and acceleration projecticle motion
[tex]m\frac{dv(t)}{dt} + kv(t) + mg = 0[/tex] You can then solve it using the traditional techniques discussed here: http://hyperphysics.phyastr.gsu.edu...h/deinhom.html Hope that helps. 


#5
Aug2109, 04:51 PM

P: 9

[tex]\frac{mdv}{mg+kv}=dt[\tex]
[tex]\int\frac{mdv}{mg+kv}=\intdt[\tex] [tex]\frac{m}{k}\ln{mg+kv}=t+c[\tex] [tex]mg+kv=\exp^\frac{k}{m}t+c[\tex] ah I cant for the life od me figure this fancy way of displaying the functions like you would on paper so I will gracefully withdraw. 


#6
Aug2209, 12:25 PM

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P: 12,064

[tex]\frac{mdv}{mg+kv}=dt[/tex] [tex]\int\frac{mdv}{mg+kv}=\intdt[/tex] [tex]\frac{m}{k}\ln{mg+kv}=t+c[/tex] [tex]mg+kv=\exp^\frac{k}{m}t+c[/tex] 


#7
Aug2209, 03:12 PM

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#8
Aug2209, 03:33 PM

P: 9

[tex]F=ma[/tex]
[tex]F=ma=(mgkv)[/tex] [tex]a=\frac{dv}{dt}[/tex] [tex]\frac{mdv}{dt}=(mgkv)[/tex] [tex]\frac{mdv}{mg+kv}=dt[/tex] [tex]\int\frac{mdv}{mg+kv}=\intdt[/tex] [tex]\frac{m}{k}\ln{mg+kv}=t+c[/tex] [tex]mg+kv=\exp^\frac{k t}{m}+c[/tex] [tex]v=C\exp^\frac{k t}{m}\frac{mg}{k}[/tex] [tex]v=v0[/tex] [tex]t=0[/tex] [tex]v0=C\exp^\frac{k}t{m}\frac{mg}{k}[/tex] [tex]v0=C\exp^0\frac{mg}{k}[/tex] [tex]c=v0\frac{mg}{k}[/tex] [tex]V=(v0\frac{mg}{k})\exp^\frac{k (0)}{m}\frac{mg}{k}[/tex] 


#9
Aug2209, 03:42 PM

P: 9

[tex]F=ma[/tex]
[tex]F=ma=(mgkv)[/tex] [tex]a=\frac{dv}{dt}[/tex] [tex]\frac{mdv}{dt}=(mgkv)[/tex] [tex]\frac{mdv}{mg+kv}=dt[/tex] [tex]\int\frac{mdv}{mg+kv}=\intdt[/tex] [tex]\frac{m}{k}\ln{mg+kv}=t+c[/tex] [tex]mg+kv=\exp^\frac{kt}{m}+c[/tex] [tex]v=C\exp^\frac{kt}{m}\frac{mg}{k}[/tex] [tex]v=v0[/tex] [tex]t=0[/tex] [tex]v0=C\exp^\frac{k(0)}{m}\frac{mg}{k}[/tex] [tex]v0=C\exp^0\frac{mg}{k}[/tex] [tex]c=v0+\frac{mg}{k}[/tex] [tex]V=(v0+\frac{mg}{k})\exp^\frac{k t}{m}\frac{mg}{k}[/tex] Velocity displacement is [tex]s=\int vdt[/tex] [tex]s=\int(v0+\frac{mg}{k})\exp^\frac{k t}{m}\frac{mg}{k}dt[/tex] [tex]s=\frac{m}{k}(v0+\frac{mg}{k})\exp^\frac{k t}{m}\frac{mg}{k}tc[/tex] [tex]s=0[/tex] [tex]t=0[/tex] [tex]0=\frac{m}{k}(v0+\frac{mg}{k})\exp^\frac{k (0)}{m}\frac{mg}{k}(0)+c[/tex] [tex]0=\frac{m}{k}(v0+\frac{mg}{k})\exp^00+c[/tex] [tex]0=\frac{m}{k}(v0+\frac{mg}{k})1+c[/tex] [tex]c=\frac{m}{k}(v0+\frac{mg}{k})[/tex] [tex]s=\frac{m}{k}(v0+\frac{mg}{k})\exp^\frac{k t}{m}\frac{mg}{k}t+\frac{m}{k}(v0\frac{mg}{k})[/tex] [tex]s=\frac{m}{k}(v0+\frac{mg}{k})(1\exp^\frac{k t}{m}\frac{mg}{k}t[/tex] Displacement 


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