Question

coffee filter lab
if an object of mass m falls under the influence of gravity and a drag force f_drag, we may write newtons second law as ma = mg - f_drag
where a is the acceleration of the object and mg is the weight of the object. the drag force has the general form: f_drag = bv^n
where the drag coefficient b is a constant that depends on the shape of the object, and v is the velocity of the object. as an object falls from rest the velocity increases until the drag force and the weight are equal in magnitude. the acceleration then becomes zero and we have the following relation: mg = b(v_t)^n
in this lab, the mass (m) will be comprised of coffee filters. we can then express the total mass in terms of the number of filters (n) and the mass of one filter (m): m = n(m)
now we can express a relationship between the number of filters and the terminal velocity. n(m)g = b(v_t)^n or n ∝ (v_t)^n
our goal in this lab is to find n for the drag force.
preliminary questions

1. drop one filter and then drop two. did the two filters seem to fall faster, slower or at the same rate? what kind of mathematical relationship do you expect will exist between the velocity of the coffee filter and the number of filters?
2. based on observations, sketch a graph of velocity vs. time for one filter as it falls to the ground. do your best to guess approximate times and velocities on your axes. also sketch a graph of acceleration vs. time.

Explanation:

Step1: Analyze the falling - rate question

When two filters are dropped instead of one, the two filters have a greater mass. According to $N(m)g = b(v_t)^n$, with $b$ being a constant related to the object's shape, as the number of filters $N$ increases, the terminal velocity $v_t$ increases. So two filters fall faster. We expect a power - law relationship between the velocity of the coffee filter and the number of filters, $N\propto(v_t)^n$.

Step2: Sketch velocity - time graph

At the start ($t = 0$), the velocity $v=0$. As the filter falls, the gravitational force causes it to accelerate and the velocity increases. As air resistance builds up, the acceleration decreases until the air resistance equals the gravitational force and the filter reaches terminal velocity. The graph starts at the origin, has a positive slope that gradually decreases until it becomes zero at terminal velocity.

Step3: Sketch acceleration - time graph

At $t = 0$, the acceleration $a = g$ (since the initial velocity is 0 and air resistance is 0). As the filter falls and the velocity increases, the air resistance increases. According to $Ma=Mg - F_{drag}$, the acceleration $a$ decreases. When terminal velocity is reached, $a = 0$. So the acceleration - time graph starts at $a = g$ and decreases exponentially to 0.

Answer:

1. The two filters fall faster. We expect a power - law relationship $N\propto(v_t)^n$ between the velocity of the coffee filter and the number of filters.
2. For the velocity - time graph: It starts at the origin $(0,0)$, has an increasing positive - slope curve that flattens out as it reaches terminal velocity. For the acceleration - time graph: It starts at $a = g$ on the $a$ - axis and decreases exponentially to 0 as time $t$ increases.