- #1
sid9221
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"Simple" Newtons laws question ?
At time t = 0 a particle of unit mass is projected vertically upward with velocity v_0 against gravity, and the resistance of the air to the particle's motion is κ times its velocity.
Show that during its flight the velocity v of the particle at time t is
[tex] v = (v_0 +\frac{g}{k}) e^{-kt} - \frac{g}{k} [/tex]
Now what I have done is work with Newtons 2nd law:
[tex] mx'' = -mκv [/tex]
[tex] v' = -κv [/tex]
solving this differential equation gives:
[tex] v=Ce^{-kt} [/tex]
Now v(0) = v_0 from question so:
C=v_0
[tex] v=v_0 e^{-kt} [/tex]
Now clearly I missing quite a bit of stuff, so where exactly am I going wrong ?
At time t = 0 a particle of unit mass is projected vertically upward with velocity v_0 against gravity, and the resistance of the air to the particle's motion is κ times its velocity.
Show that during its flight the velocity v of the particle at time t is
[tex] v = (v_0 +\frac{g}{k}) e^{-kt} - \frac{g}{k} [/tex]
Now what I have done is work with Newtons 2nd law:
[tex] mx'' = -mκv [/tex]
[tex] v' = -κv [/tex]
solving this differential equation gives:
[tex] v=Ce^{-kt} [/tex]
Now v(0) = v_0 from question so:
C=v_0
[tex] v=v_0 e^{-kt} [/tex]
Now clearly I missing quite a bit of stuff, so where exactly am I going wrong ?