Question

我正在尝试构建一个函数，该函数将计算给定时间步长的物体的牛顿引力定律（G * M / r ^ 2），同时能够改变该点的位置和速度。

作为一个例子，我将尝试展示我所做的设置初始速度和点的起始位置（但是我会在某些时候需要能够实时显示该点并移动它循环正在运行。）

我写了两个试图证明我遇到麻烦的例子。

如果仔细观察图表，您可能会注意到这两行不重叠（即使所有起始参数都相同）。

此图显示所有起始参数= 0 的时间

打印输出：

概念1 （计算间隔= 1秒）：

v:9.82  pos:9.82
v:19.64 pos:29.46
v:29.46 pos:58.93
v:39.28 pos:98.22

概念2 （计算间隔= 1秒）：

v:0.0   pos:0.0  
v:9.82  pos:4.91
v:19.64 pos:19.64
v:29.46 pos:44.20

根据此online calculator，它表示Concept1不正确，而Concept2非常接近实际。

以下代码绘制了两个示例。我添加了两个变量，可以设置initial_velocity和initial_position（请参阅下面的其他两个情节图片了解详情）。

import numpy as np
import matplotlib.pyplot as plt

G = 6.67408 * 10 ** -11  # Newton meters^2/kg^2
Me = 5.9736 * 10 ** 24  # mass of earth in kg
Re = 6.371 * 10 ** 6  # radius of earth in meters

def get_g(mass, distance, seconds):
    _g = G * mass / distance ** 2
    _d = (_g * (seconds ** 2)) / 2
    _v = _g * seconds
    return _v, _d, _g

def Concept1(seconds=1, timestep=0.1):
    plt_x = np.linspace(0, seconds, (seconds/timestep))
    plt_y = np.zeros_like(plt_x)

    """
    Simple concept demonstrating values that are very close to actual.

    Time =  0.1  vel =  0.9822280588290989  pos =  0.049111402941454954  _g =  9.822280588290988
    Time =  0.2  vel =  1.9644561176581978  pos =  0.19644561176581982   _g =  9.822280588290988
    Time =  0.3  vel =  2.946684176487297   pos =  0.44200262647309463   _g =  9.822280588290988
    Time =  0.4  vel =  3.9289122353163957  pos =  0.7857824470632793    _g =  9.822280588290988
    """

    for _i in range(len(plt_y)):
        t = (_i+1)
        _v, _d, _g = get_g(Me, Re, t*timestep)

        plt_y[_i] = _d
        #print("Time = ", t*timestep, " vel = ", _v, " pos = ", _d, " _g = ", _g)

    return plt_x,plt_y

def Concept2(seconds=1, timestep=0.1, initial_velocity=0, initial_position=0):

    plt_x = np.linspace(0, seconds, seconds/timestep)
    plt_y = np.zeros_like(plt_x, dtype=np.float32)

    """
    Dynamic concept showing variable velocity and starting position (output is not accurate)

    Time =  0.1  vel =  0.9822280588290989  pos =  0.0982228058829099  _g =  9.822280588290988
    Time =  0.1  vel =  1.9644561176581978  pos =  0.2946684176487297  _g =  9.822280588290988
    Time =  0.1  vel =  2.9466841764872966  pos =  0.5893368352974594  _g =  9.822280588290988
    Time =  0.1  vel =  3.9289122353163957  pos =  0.982228058829099   _g =  9.822280588290988
    """

    vel = initial_velocity * timestep
    pos = initial_position

    for _i in range(len(plt_y)):
        t = timestep
        _v, _d, _g = get_g(Me, Re, t)

        vel += _g * timestep
        pos += vel * timestep

        plt_y[_i] = pos
        #print("Time = ", t, " vel = ", vel, " pos = ", pos, " _g = ", _g)

    return plt_x, plt_y

def Concept3(seconds=1, timestep=0.1, initial_velocity=0, initial_position=0):

    plt_x = np.linspace(0, seconds, seconds/timestep)
    plt_y = np.zeros_like(plt_x, dtype=np.float32)

    """
    Dynamic concept showing variable velocity and starting position (output is accurate)

    Time =  0.1  vel =  0.982228058829099   pos =  0.049111402941454954  _g =  9.822280588290988
    Time =  0.1  vel =  1.964456117658198   pos =  0.19644561176581982   _g =  9.822280588290988
    Time =  0.1  vel =  2.9466841764872975  pos =  0.44200262647309463   _g =  9.822280588290988
    Time =  0.1  vel =  3.928912235316396   pos =  0.7857824470632793    _g =  9.822280588290988
    """

    vel = initial_velocity * timestep
    pos = initial_position
    gforce = 0

    for _i in range(len(plt_y)):
        t = timestep

        _v, _d, _g = get_g(Me, Re, t)

        vel += (gforce + _g)*timestep**2/2
        pos += vel
        gforce = _g

        plt_y[_i] = pos
        #print("Time = ", t, " vel = ", (vel + _d)/timestep, " pos = ", pos, " _g = ", gforce)

    return plt_x, plt_y


seconds=1
timestep=0.1
initial_velocity = 0
initial_position = 0
x1, y1 = Concept1(seconds=seconds, timestep=timestep)
x2, y2 = Concept2(seconds=seconds, timestep=timestep, initial_velocity=initial_velocity, initial_position=initial_position)
x3, y3 = Concept3(seconds=seconds, timestep=timestep, initial_velocity=initial_velocity, initial_position=initial_position)

plt.xlabel("Time (Seconds)")
plt.ylabel("Distance")
plt.plot(x1, y1,'g', label='Concept 1 (gauge)')
plt.plot(x2, y2,'b', label='Concept 2')
plt.plot(x3, y3,'r', label='Concept 3')
plt.legend(loc='upper left')
plt.show()

我对物理学比较陌生（这是一种学习经历）。我一直想知道我的方法是否可能不正确。如果是这样，我真的想了解更多如何实现这一目标。我们很乐意留意所有建议。谢谢你！

显示initial_velocity = 2000 的情节

显示initial_position = 100000 的情节

[UPDATE]

在代码中添加了Concept3。

我“认为”它通过在每个循环迭代中将当前gforce值存储在变量中来实现我所需要的。

然后我将它添加到当前的gforce值并将其除以2。之后我更新当前的速度。

这是进行更改后的输出（秒= 1次步长= 0.1 ）：

在计算牛顿万有引力定律时如何控制点的速度和位置？

0 个答案: