美賽數模論文之公式寫作

由假設得到公式

1．We assume laminar flow and use Bernoulli's equation:（由假設得到的公式）

公式

Where

符号解釋

According to the assumptions, at every junction we have （由于假設）

公式

由原因得到公式

2．Because our field is flat, we have公式, so the height of our source relative to our sprinklers does not affect the exit speed v2 （由原因得到的公式）；

公式

Since the fluid is incompressible(由于液體是不可壓縮的), we have

公式

Where

公式

用原來的公式推出公式

3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到）

公式

11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：

公式

12．Therefore, from (2),(3),(5), we have the ith junction（由前幾個公式得）

公式

Putting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have

公式

Putting these into (1) ,we get（把這些公式代入1中）

公式

Which means that the

Commonly, h is about

From these equations, （從這個公式中我們知道）we know that ………

引出限制條件

4．Using pressure and discharge data from Rain Bird 結果,

We find the attenuation factor （得到衰減因子，常數，系數） to be

公式

計算結果

6．To find the new pressure ,we use the ( 0 0),which states that the volume of water flowing in equals the volume of water flowing out : （為了找到新值，我們用什麼方程）

公式

Where

() is ;;

7．Solving for VN we obtain (公式的解)

公式

Where n is the …..

8．We have the following differential equations for speeds in the x- and y- directions:

公式

Whose solutions are （解）

公式

9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:

公式

根據原有公式

10．We apply the law of conservation of energy（根據能量守恒定律）. The work done by the forces is

公式

The decrease in potential energy is （勢能的減少）

公式

The increase in kinetic energy is (動能的增加)

公式

Drug acts directly against velocity, so the acceleration vector from drag can be found Newton's law F=ma as : (牛頓第二定律)

Where a is the acceleration vector and m is mass

Using the Newton's Second Law, we have that F/m=a and

公式

So that

公式

Setting the two expressions for t1/t2 equal and cross-multiplying gives

公式

22．We approximate the binomial distribution of contenders with a normal distribution:

公式

Where x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives

公式

As an analytic approximation to . for k=1, we get B=c

26．Integrating, （使結合）we get PVT=constant, where

公式

The main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so

23．According to First Law of Thermodynamics, we get

公式

Where ( ) . we also then have

公式

Where P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:

公式

Where

對公式變形

13．Define A=nlw to be the ( )（定義）; rearranging (1) produces （将公式變形得到）

公式

We maximize E for each layer, subject to the constraint (2). The calculations are easier if we minimize 1/E.（為了得到最大值，求他倒數的最小值） Neglecting constant factors （忽略常數）, we minimize

公式

使服從限制條件

14．Subject to the constraint （使服從限制條件）

公式

Where B is constant defined in (2). However, as long as we are obeying this constraint, we can write （根據限制條件我們得到）

公式

And thus f depends only on h , the function f is minimized at （求最小值）

公式

At this value of h, the constraint reduces to

公式

結果說明

15．This implies（暗示） that the harmonic mean of l and w should be

公式

So , in the optimal situation. ………

5．This value shows very little loss due to friction.（結果說明） The escape speed with friction is

公式

16． We use a similar process to find the position of the droplet, resulting in

公式

With t=0.0001 s, error from the approximation is virtually zero.

17．We calculated its trajectory(軌道) using

公式

18．For that case, using the same expansion for e as above,

公式

19．Solving for t and equating it to the earlier expression for t, we get

公式

20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is

公式

As v=…, this equation becomes singular (單數的).

由語句得到公式

21．The revenue generated by the flight is

公式

24．Then we have

公式

We differentiate the ideal-gas state equation

公式

Getting

公式

25．We eliminate dT from the last two equations to get （排除因素得到）

公式

22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations

公式

Where P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）

公式