Differential Eq falling object + friction

[iloveannaw]

Homework Statement

A point mass m falls from rest through a height h. The frictional force is given by $$-\gamma \dot{z}$$ and gravity by $$-mg$$.

Give the 'equation of motion' (differential equation) for the height z(t).

The Attempt at a Solution

$$\ddot{z} = -g - \frac{\gamma}{m} \dot{z}$$

I thought about integrating then rearranging:

$$\Rightarrow \dot{z} = -gt - \frac{\gamma}{m}z +c_{1}$$

$$\Rightarrow z(t) = - \frac{m}{\gamma} (\dot{z} + gt + c_{1})$$

The question the asks what kind of differentional eq. this is and asks the student to make the following substitution:

$$\dot{z}(t) = v(t)$$

and asks what kind of equation it is now! Well I haven't got a clue what its is on about. I assume something like $$s = ut +\frac{1}{2}at^{2}$$ should come out. Have to hand this in tomorrow so please help!

 

[tiny-tim]

hi iloveannaw!

The question the asks what kind of differentional eq. this is and asks the student to make the following substitution …

it means substitute in the original equation (the one beginning z'')

 

[iloveannaw]

thanks, so you think I should start by working from $$\frac{dv}{dt} = -g -\frac{\gamma}{m}v$$ ?

I have already done that, but the question is quite clear it asks for the diff. eq. in terms of $$\dot{z}(t)$$ and then asks the student to make substitution. And it also asks for type of differential equation before and after.

 

[tiny-tim]

that is the differential equation in terms of z' (your other one was in terms of z' and z) …

and you should be able to solve it ​