Question

Obtain the differential equation for the velocity \(v\) of a body of mass \(m\) falling vertically downward through a medium offering a resistance proportional to the square of the instantaneous velocity.

Short answer

Answer: \(\frac{dv}{dt} = g - \frac{k}{m}v^2\)

Step-by-step solution

1. Identify the forces acting on the body

The body falling vertically downwards through a medium experiences two forces: the gravitational force (F_g) acting downwards and the resistive force (F_r) acting upwards. The gravitational force is given by \(F_g=mg\), where m is the mass of the body and g is the acceleration due to gravity. The resistive force is given by \(F_r=kv^2\), where k is a positive constant of proportionality and v is the instantaneous velocity of the body.

2. Apply Newton's second law of motion

According to Newton's second law, the net force acting on the body is equal to the product of its mass and acceleration: \(F_{net}=ma\). In this case, the acceleration (a) will be the derivative of the velocity with respect to time, that is, \(\frac{dv}{dt}\). As the body is moving downwards, we need to consider the direction of the forces. Let's take the downward direction as positive. So, the net force acting on the body will be the difference between the gravitational force and the resistive force: \(F_{net} = F_g - F_r = mg - kv^2\).

3. Create the differential equation

Now, we can substitute the expression for the net force in terms of m, g, k, and v into Newton's second law equation: \(ma=mg-kv^2\). We also know that \(a=\frac{dv}{dt}\); therefore: \(m\frac{dv}{dt}=mg-kv^2\). This is a first-order nonlinear ordinary differential equation for the velocity v in terms of time.

4. Final form of the differential equation

After obtaining the initial form of the differential equation, we can rewrite it as: \(\frac{dv}{dt} = g - \frac{k}{m}v^2\). This is the final form of our differential equation for the velocity of a body falling vertically downwards through a medium offering resistance proportional to the square of the instantaneous velocity.