第三次做业

《计算物理》第一章课后习题3

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一. 课题
It is often the case that the frictional force on an object will increase as object moves faster. A fortunate example of this is a parachutist; the role of the parachute is to produce a frictional force due to air drag, which is narmally larger than would normally be the case without the parachute. Here we consider a very simple examle in which the frictional force depends on the velocity. Assume that the velocity of a object obeys the equation of the form

$$dv/dt=a-bv$$
Where a and b are constants. You could think of a as coming from an applied force, such as gravity, while b arises from friction. Note that the frictional force is negative(we asume that b>0), so that it opposes the motion. Use the Euler method to solve for v as a function of time. A convenient choice of parameters is a=10 and b=1. You should find that v apporaches a as a constant value at long times; This is called the terminal velocity.

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二.实验目的

1 用欧拉法拟合出速度随时间的变化，并用mathplotile在python上绘出
2 该速度微分方程可以轻松算出解析解，同样地绘出解析解的速度随时间变化的曲线
3 观察收尾速度
4 改变a与b的设定值，观察变化

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三.源代码

*本有四段代码，但区别是改变a与b的设定值， 故不重复粘贴

   @author: 钟伟
"""
import numpy as np
import matplotlib.pyplot as plt
a = 10  
b = 1
"""设置初始条件"""
vi = int(input("please input the initial velovity:")) 
"""输入初始速度"""
t = 0
delt_t = 0.1
process_v = vi
"""设定两个列表储存数据"""
list_v=[process_v,] 
list_t=[t,]
"""进行计算，并将计算结果储存在列表中"""
while t<4:
    process_v = process_v + (a-b*process_v)*delt_t 
    """欧拉法计算"""
    t=t+delt_t
    list_v.append(process_v)
    list_t.append(t)
"""解析解的方程由积分法求出"""    
c=a-b*vi
t=np.linspace(0,4,1000)
v=(a-c*np.exp(-b*t))/b
"""画出图像"""
plt.figure(figsize=(50,25))
plt.plot(list_t,list_v,label="$v(t)$",marker='s')
plt.plot(t,v,linewidth=2,color="yellow")
plt.xlabel("Time(s)")
plt.ylabel("velocity(m/s)")
plt.title("velocity variation")
plt.ylim(0,11)
plt.legend()
plt.show()

创建时间
Sep 28 15:46:29 2017

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三.调试及结果

*均为v=1 a=10 模拟业余跳伞刚一跳就迫不及待打开伞 重力加速度为10m/s

1.当b=1时 有接近对数曲线的图像。时间越大拟合的越好。可认为在一定时间后达到了收尾速度。收尾速度为大约10m/s， 人可以承受。

2.当b=0.1 几乎是自由落体，相当于无伞降落，是很危险的

3.当b=5或10 很快收尾 如果降落伞能这样，那么低空跳伞就好玩了。。

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感谢观看！！！