 #1
torito_verdejo
 20
 4
 Homework Statement:

A point mass ##m## is dropped from a hight ##z##. Its motion is subject to gravity force and a friction force ##F_f=\lambda \dot{z}##.
Write the equations of motion for this system.
 Relevant Equations:
 $$F_f=\lambda \dot{z}$$
I'm stack at the very beginning. If I use Newton's second law to find acceleration and integrate until I find the position, I must face
$$v(t) = \int_0^t g\frac{\lambda v}{m} dt'=gt\frac{\lambda }{m} \int_0^t\frac{\partial z}{\partial t}dt$$
But this last term feels pretty weird. I don't know how to interpret it, even though
$$\frac{\partial z}{\partial t}dt = dz$$
If I extract velocity from friction, we get
$$v(t)=\frac{m}{\lambda}(ga(t))$$
which once again defines a function by its derivative. Should I be solving a differential equation?
In any case, I face two possible ways and I wonder which one, if any, is the correct one. Thank you in advance for the help.
$$v(t) = \int_0^t g\frac{\lambda v}{m} dt'=gt\frac{\lambda }{m} \int_0^t\frac{\partial z}{\partial t}dt$$
But this last term feels pretty weird. I don't know how to interpret it, even though
$$\frac{\partial z}{\partial t}dt = dz$$
If I extract velocity from friction, we get
$$v(t)=\frac{m}{\lambda}(ga(t))$$
which once again defines a function by its derivative. Should I be solving a differential equation?
In any case, I face two possible ways and I wonder which one, if any, is the correct one. Thank you in advance for the help.
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