Free Fall

In the free fall problem, an object of mass m, at a initial height z0, falls under the effect of a gravitational acceleration g, with a initial velocity v0. At a time instant t, the height and velocity are z and v.

The objective here is to determine some dimensionless relations between those quantities.

We start by importing the necessary classes from nodimo:

from nodimo import Q, Model

The dependent quantity will be z and there are three quantities that are interesting to set as scaling: g, z0 and v0. The quantities v and t are not suitable to be used as scaling, because they are not parameters of the problem. And it’s needless to say that m can’t be part of our model, but it was added to show how nodimo handles these situations.

z = Q('z', L=1, dependent=True)
m = Q('m', M=1)
v = Q('v', L=1, T=-1)
g = Q('g', L=1, T=-2, scaling=True)
t = Q('t', T=1)
z0 = Q('z_0', L=1, scaling=True)
v0 = Q('v_0', L=1, T=-1, scaling=True)

Now, to create all possible dimensionless models using the quantities defined above:

model = Model(z, m, v, g, t, z0, v0)

NodimoWarning: Dimensionally irrelevant quantities (m)

As expected, the quantity m is irrelevant to the model and is, therefore, discarded.

To display all possible relations, use the show method:

model.show()

$$\displaystyle z = f\left(v,\ g,\ t,\ z_{0},\ v_{0}\right)$$

---

$\displaystyle \mathtt{\text{Scaling group 1 }}\left(g,\ z_{0}\right)$

$$\displaystyle \frac{z}{z_{0}} = \Phi_{1}\left(\frac{v}{{g}^{\frac{1}{2}} {z_{0}}^{\frac{1}{2}}},\ \frac{t {g}^{\frac{1}{2}}}{{z_{0}}^{\frac{1}{2}}},\ \frac{v_{0}}{{g}^{\frac{1}{2}} {z_{0}}^{\frac{1}{2}}}\right)$$

---

$\displaystyle \mathtt{\text{Scaling group 2 }}\left(g,\ v_{0}\right)$

$$\displaystyle \frac{z g}{{v_{0}}^{2}} = \Phi_{2}\left(\frac{v}{v_{0}},\ \frac{t g}{v_{0}},\ \frac{z_{0} g}{{v_{0}}^{2}}\right)$$

---

$\displaystyle \mathtt{\text{Scaling group 3 }}\left(z_{0},\ v_{0}\right)$

$$\displaystyle \frac{z}{z_{0}} = \Phi_{3}\left(\frac{v}{v_{0}},\ \frac{t v_{0}}{z_{0}},\ \frac{g z_{0}}{{v_{0}}^{2}}\right)$$

In this way, you are able to choose the dimensionless relation that better represents the model for your purpose.