Exercise 1.3: Air resistance

The phenomenological form of the velocity-dependent force of the resistance of the air is given by

$$F_d=kv$$

where $k$ is a parameter that depends on the properties of the medium and the shape of the object. Since $F$ increases as $v$ increases, there is a limiting terminal velocity at which $F_d=F_g=mg$ and the acceleration vanishes:

$$kv_t=mg \Rightarrow v_t=\frac{mg}{k}$$

In terms of the terminal speed, the force $F_d$ can be rewritten as

$$F_d=mg(\frac{v}{v_t}).$$

Hence, the net force on a falling object is:

$$F=mg(1-\frac{v}{v_t})$$

1. Sometimes, the force $F_d$ can vary with the square of the velocity
   
   $$F_d=k_2 v^2.$$
   
   
   Derive the net force on a falling object using this expression, in units of the terminal velocity $v_t$

2. Compute the speed at which a pebble of mass $m=10^{-2}kg$ reaches the ground if it's dropped from rest ate $y_0=50m$. Compare this speed to that of a freely falling object under the same conditions. Assume that the drag force is proportional to $v^2$ and the terminal speed is $v_t=30m/s$

3. Supouse that an object is thrown vertically upward with initial velocity $v_0$. If we neglect air resistance, we know that the maximum height reached by the object is $v_0^2/2g$, and its velocity upon return to the earth equals $v_0$, the time of ascent equals the time of descent, and the total time in the air is $v_0/g$. Before performing a numerical simulation, give a simple qualitative analysis of the problem when it is affected by the resistance of the air. Then, perform, the numerical calculation assuming $F_d \sim v^2$ with a terminal speed $v_t=30m/s$. Suggestions: Choose $v>0$ when it's pointing upward, and $v<0$ when it's pointing toward the earth.