Euler's method for motion

What you will be able to do

After this lesson you can

• write a motion problem as a rate of change,
• follow it in small steps with Euler's method, and
• compare the steps with the exact answer, and with solve_ivp of SciPy.

Rates of change

In physics, most laws tell you how fast something changes, not how big it is. Newton's second law says that the acceleration is the force divided by the mass. The acceleration is the rate of change of the speed. So the law is an equation about how the speed changes:

Such an equation is a differential equation. To find the speed itself at a later time, we need to solve it. For simple forces, you do it with calculus. For many forces, there is no formula for the answer, and the computer solves it, in a way that follows the story of the motion step by step.

Euler's method

Imagine that you are riding a bicycle with your eyes closed. You know your speed now. In a short time $dt$, you will go about $v \cdot dt$ forward. And you know how your speed is changing, so in the same short time your speed will change by $a \cdot dt$. After the step, you look again, and repeat.

The rule, for a position $x$ and a speed $v$:

This is Euler's method. It is a loop that moves forward in time in small steps.

Predict first. A stone falls for 3 seconds. Will Euler's answer for the distance be exact, too big or too small?

The output:

• The speed after 3 s is exactly 29.4 m/s, as the exact formula $gt$ says. (For a constant acceleration, Euler's method is exact for the speed.)
• The distance is 42.63 m, but the exact distance $\tfrac{1}{2}gt^2$ is 44.1 m. Euler's answer is too small, because in each step it uses the speed at the start of the step, and the real speed is growing during the step.

Euler's method is not exact. How good is it?

Air drag

A falling ball feels the force of the air, which grows with the speed. In a simple model, it gives the acceleration $-kv$, so

with $g = 9.8$ m/s$^2$ and $k = 0.5$ per second. This equation has the exact solution $v(t) = \dfrac{g}{k}\left(1 - e^{-kt}\right)$. At first, the ball speeds up as in free fall. As the speed grows, the drag grows, and the speed approaches the terminal speed, where $g - kv = 0$, so $v = g/k$.

How accurate is it?

Predict first. When the step is halved, by how much will the error change?

The picture shows the same thing. The orange line (a step of 0.1) follows the exact black curve closely. The blue line (a step of 1) leaves the curve: it overshoots, because each step uses the rate of change at the start of the step, and the rate is falling.

Both settle at the terminal speed 9.8 / k, where the drag balances the weight and the slope becomes 0.

SciPy does the work

For real work, use solve_ivp ("solve an initial value problem") from SciPy. It chooses the steps by itself, small where the solution changes fast, big where it is smooth, and uses a much better method than Euler's.

So Euler's method is for understanding, and solve_ivp is for work. Understanding Euler's method tells you what solve_ivp does behind the scenes: it follows the rates of change, step by step.

Common mistakes

• Updating the speed before using it for the position. In the basic method, use the old values.
• A step that is too big. The curve is wrong. Halve the step and see if the answer changes.
• Forgetting that rate gets t first and y second.
• Returning a number where a list is needed. The function returns a list, one rate for each quantity.
• Believing a solution without a check. Compare with a case that you know.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Euler's method. You write the method for any rate of change.
2. Build. Open the problem A cooling cup and the problem Terminal speed.

Remember these

• A differential equation gives the rate of change. Euler's method moves forward in small steps: new value = old value + rate $\times\ dt$.
• Euler is first order: half the step gives half the error.
• With air drag $-kv$, the speed approaches $g/k$.
• solve_ivp(rate, (t0, t1), start) solves it with steps that it chooses. rate(t, y) returns a list of rates.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A ball falls with $v = 10$ m/s, and the acceleration is 2 m/s$^2$. What is $v$ after a step of 0.5 s?

2. The error of an Euler solution is 0.08 with $dt = 0.2$. What do you expect with $dt = 0.1$?

3. What is the terminal speed for $\dfrac{dv}{dt} = 9.8 - 0.2v$?

4. What does the function rate(t, y) return?