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- 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

$$v(t)=\frac{m}{\lambda}(g-a(t))$$

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

$$v(t)=\frac{m}{\lambda}(g-a(t))$$

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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