Numerical ODE Solving

The code provided uses Euler's method to numerically solve the ordinary differential equation (ODE) describing free-fall motion with air resistance.

1. Euler's Method Function

• euler_method takes a function f (which represents the ODE), initial value y0, time range t0 to tf, and step size h. It computes the numerical solution using the Euler method.

2. ODE Definition

• The ODE for free-fall with air resistance is:

  [ \frac{dv}{dt} = g - \frac{k}{m}v ]

  where:

  • ( g ) is gravity (9.81 m/s²),
  • ( k ) is the air resistance constant (0.1),
  • ( m ) is the mass of the object (70 kg),
  • ( v ) is the velocity of the object.

3. Simulation Setup

• The initial velocity ( v_0 = 0 ) m/s (starting from rest).
• The time range is from ( t_0 = 0 ) to ( t_f = 10 ) seconds with a time step ( h = 0.01 ) seconds.

4. Plot

• The velocity over time is plotted using the results from Euler’s method.

Running this code will produce a graph of the velocity as a function of time, showing how air resistance affects the velocity of an object during free fall.