Exercise 04 Euler Method

（15级物院弘毅 陈亦林 2015301020152）

计算物理作业 Quastion1-3跳伞员问题

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Abstract : In this exercise, with Matplotlib and Cmd Markdown, I will able to solve the problem of parachutist in Question1-3, by using Euler Method.
Keyword: Matplotlib, Cmd Markdown, Problem of parachutist, Euler Method

I. Question

Consider a example in which the frictional force of parachutist depends on the velocity. Assume that the velocity of him obeys an equation of the form.

$$\frac{\mathrm{d} v }{\mathrm{d} t}= a-bv$$
where a and b are constants.
Using Euler Method to solve it for v as a function of time. We can choose$a=10$ and $b=1$. You should find that v approaches a constant value at long times; this is called the terminal velocity.

II. Euler Method

First, we can put $v$ in Taylor Expansion.

$$v(\ t)=v(\ t-0)=v(0)+\frac{dv(t)}{ dt}\Delta t+\frac{1}{2} \frac{d^2v(t)} {d^2t}(\Delta t)^2+···$$
We just reserve the expansion of the first order, and put it in the function.
$$\frac{dv(t)}{dt}≈ \frac{v(t+\Delta t)-v(t)}{\Delta t}$$
Then $$\frac{v(t+\Delta t)-v(t)}{\Delta t}=a-bv$$
Finally, we get $$v(t+\Delta t)≈v(t)+(a-bv)\Delta t$$

III. Results & Code

We regard the initial velocity as ${v}_0$, $a=10$ and $b=1$.

Euler Mthod Code

1. When ${v}_0=0$, we have v-t figure

2. When ${v}_0=10$, we have v-t figure

3. When ${v}_0=20$, we have v-t figure

IV. Conclusion

( In this model, we have $a=10$ and $b=1$. )

1.If the initial velocity is lower than 10, as ${v}_0=0$.
The gravity is larger than the resistance. And the speed will increase, accompanied by the resistance. This force will become smaller, since the acceleration will also decrease, but the direction is vertical downward, so the image on the slope becomes smaller, until the speed is reduced to 10. The gravity is equal to the resistance. Force will become 0 as well as the acceleration become 0.

2.If the speed is equal to 10, as ${v}_0=10$.
The gravity is equal to the resistance. The speed will remain 10. $v$ approaches a constant value 10 at all time.

3.If the speed is higher than 10, as ${v}_0=20$.
The gravity is larger than the resistance. And the speed will decrease, accompanied by a reduced resistance, This force will become smaller, since the acceleration will also decrease, but the direction is vertical, keep speed, so the image on the slope becomes smaller, until the speed is reduced to 10. The gravity is equal to the resistance. Force will become 0 as well as the acceleration become 0. That is the reason why $v$ approaches a constant value at long time.

4.All in all, When the speed of parachute approaches 10, the gravity is equal to the resistance. Force will become 0 as well as the acceleration become 0. That is the reason why $v$ approaches a constant value at long time.