Exercise 1

Aristotle’s (wrong) law is given by:

\[ \dot{x}(t) = \frac{F(t)}{m} \]

and after integration:

\[ x(t) = \int \frac{F(t)}{m} \mathrm{d}t \]

Let \(F(t) = 2t^2\): \[ x(t) = \frac{1}{m}\int 2t^2 \mathrm{d}t = \frac{2}{3m} t^3 + C \]

The constant of integration can be determined from the initial condition \(x(0) = \pi\): \[ x(0) = \pi = \frac{2}{3m} 0^3 + C \Rightarrow C = \pi \]

Hence the function of position \(x(t)\) is given by:
\[ x(t) = \frac{2}{3m} t^3 + \pi \]