To determine the velocity as a function of time for a body falling in a medium with resistance proportional to the square of the velocity, we can set up a differential equation based on Newton's second law.
Let's denote the velocity as v(t) and the proportionality constant as k. According to the problem statement, the resistance is proportional to the square of the velocity, so the resistance force can be written as F_res = -k * v^2.
The net force acting on the body is given by F_net = m * a, where m is the mass of the body. In this case, the mass is given as 8 grams, which can be converted to kg (1 gram = 0.001 kg).
Since the body is falling vertically, the net force can be expressed as the gravitational force minus the resistance force:
F_net = mg - k * v^2
Using Newton's second law, F_net = m * a, we have:
m * a = mg - k * v^2
Substituting the mass value and rearranging the equation, we get:
a = g - (k/m) * v^2
But we also know that acceleration is the derivative of velocity with respect to time (a = dv/dt), so we have:
dv/dt = g - (k/m) * v^2
This is a separable differential equation. We can rearrange it to isolate the variables:
dv / (g - (k/m) * v^2) = dt
Now we can integrate both sides. Integrating the left side involves using partial fractions or trigonometric substitutions, but it can become quite involved. To simplify the equation, we can make use of the limiting velocity mentioned in the problem.
When the body reaches its limiting velocity, the net force becomes zero. Thus, we have:
F_net = mg - k * v^2 = 0
Solving for v, we get:
v = sqrt(g / (k/m))
Substituting the given values, we find:
v = sqrt(9.8 / (k/0.008))
v = sqrt(9.8 * 0.008 / k)
We are given that the limiting velocity is 4 cm/s, so we can set up an equation:
4 = sqrt(9.8 * 0.008 / k)
Squaring both sides and rearranging, we find:
k = 9.8 * 0.008 / 16
k = 0.0049
Now we have the value of k, we can proceed with the integration.
Integrating the equation:
dv / (g - (k/m) * v^2) = dt
involves a substitution and trigonometric functions, resulting in:
arctan(sqrt(k/m) * v) = (sqrt(k/m) * g) * t + C
where C is the constant of integration.
Finally, solving for v(t), we have:
v(t) = sqrt(tan((sqrt(k/m) * g) * t + C) * (m/k))
Keep in mind that the integration constant C can be determined by considering the initial condition. Since the body starts from rest, we have v(0) = 0, which allows us to solve for C.
With the specific values provided in the problem, you can substitute the appropriate values and solve for the velocity as a function of time using the derived equation.