美赛数模论文之公式写作

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

, |4 ~3 c  {+ p$ i+ w+ m公式
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Where

: D3 N* N: u, a/ J% _5 J) E符号解释

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

* u0 v. A7 |: M3 p2 L' r公式

+ V2 s' I1 t$ @9 i2 ~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）；
6 ]' d$ S! g+ ~4 m# ^, P
! Z& w, O( S6 C0 v) {* y1 A公式
+ S4 Z+ i  F4 l
0 n8 H3 h) q3 Q6 KSince the fluid is incompressible(由于液体是不可压缩的), we have

公式

Where

# M6 w' R  K' C# ]公式

用原来的公式推出公式
3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到）
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公式
: }3 C! C, U6 _+ k& U
; V+ T: z9 _# `11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：
9 x, Y- ^5 C, X* H7 H2 n( T
% k3 {6 R+ y) J2 P0 r3 v: D公式
" K, x+ f+ b$ y; y* Y) q% p
12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）
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: y2 D3 M% C% Y) `# I公式

: Q4 w! g$ z9 g' fPutting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have
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公式

" \$ t% R' l! {7 c0 rPutting these into (1) ,we get（把这些公式代入1中）
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公式
' K  |6 w( O/ \, I2 ^+ U
Which means that the

Commonly, h is about

From these equations, （从这个公式中我们知道）we know that ………

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$ d0 M" z0 K& k% |' ]引出约束条件
4．Using pressure and discharge data from Rain Bird 结果,
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We find the attenuation factor （得到衰减因子，常数，系数） to be
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公式
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4 |" \4 S9 J/ G1 V& R; W4 f计算结果
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 : （为了找到新值，我们用什么方程）

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2 B! v  @$ }7 z& t1 z; a, K4 V: j  }2 m6 ZWhere
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() is ;;
! Z7 `# j3 X; O% J: Z7 b  X; z
  W) U7 N5 u, K- I; `8 k7．Solving for VN we obtain (公式的解)

" A# \6 D# Y# ~公式
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" `$ r! t, @" L* T/ `Where n is the …..
9 B  U3 w; [- |% A# M6 N2 ?- a2 C
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7 |4 X" Q; z. U9 I# w1 ?8．We have the following differential equations for speeds in the x- and y- directions:

) v5 s+ o& E' D" h公式

1 }( f2 `& w9 pWhose solutions are （解）

公式
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) F1 j$ N% Y& u2 K, j9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:

公式

根据原有公式
0 E  D/ L  V/ @10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is

公式
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: p; \( r! w/ ^* k9 k: {6 AThe decrease in potential energy is （势能的减少）

) R/ D& y4 u, a! z  I公式
: _" ^3 H9 `. O3 s* F
The increase in kinetic energy is (动能的增加)

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Drug acts directly against velocity, so the acceleration vector from drag can be found Newton's law F=ma as : (牛顿第二定律)
/ y. [% h$ \" V7 h1 @2 c
$ c# z+ w0 m$ L# G$ F) Y* tWhere a is the acceleration vector and m is mass
1 m; {& s' g  S0 u$ X  C
, u, x, {( W+ }/ _0 z) S* l3 j 

. D1 {) t3 s. q' E& p2 XUsing the Newton's Second Law, we have that F/m=a and

! A7 t: H  [8 w8 x- `& D5 ^公式
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' l2 x, }6 r: g- ~( }0 x5 JSo that

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

5 N7 \) P+ k: r$ E# m# N公式

3 Q5 q+ P" f" p0 X22．We approximate the binomial distribution of contenders with a normal distribution:
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公式

# y" B) M+ a) B- u) rWhere x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives

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+ Z+ }% `* C6 E& z" B; ^
As an analytic approximation to . for k=1, we get B=c

 
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26．Integrating, （使结合）we get PVT=constant, where
) X  m* X9 Z* g7 D$ c
9 B2 S: D2 T1 i, n# E公式

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

8 c: F; m; v# v! K 
7 i# t4 h6 T% f  i) Q
# H1 V3 e/ {8 \) F9 ~23．According to First Law of Thermodynamics, we get

公式

2 g4 }* u9 k8 B3 D9 IWhere ( ) . we also then have

6 Y& k) t: V9 q9 {) V) l- w公式

0 }+ h# U4 v9 eWhere P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:

公式
0 a% P) {% J" h9 M4 H3 A& a
: p% a6 |! w1 h" R( nWhere

1 m/ H% r3 J! |8 C0 m 

对公式变形
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到）

1 t% z& @" ]5 E, _' V# }: w公式

We maximize E for each layer, subject to the constraint (2). The calculations are easier if we minimize 1/E.（为了得到最大值，求他倒数的最小值） Neglecting constant factors （忽略常数）, we minimize
- k0 t0 `" S- P/ o7 j; ~: a
+ z+ Y9 K2 y. `( b公式

1 {: C4 e" ]1 V3 s2 G5 k9 y1 A6 u( j使服从约束条件
14．Subject to the constraint （使服从约束条件）
, F0 l  f+ o& \0 z: v) A. X
7 ^# F7 t5 s0 O5 u公式

Where B is constant defined in (2). However, as long as we are obeying this constraint, we can write （根据约束条件我们得到）

# p* o9 {. ]8 R( p- p公式

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

公式
6 m. C( V& K! S8 h& K+ b/ B
At this value of h, the constraint reduces to

( B+ a  {' q4 u2 [+ P公式

0 V: _3 Y$ z# [1 c% [结果说明
& \0 \" M/ G* g. e; Q15．This implies（暗示） that the harmonic mean of l and w should be

公式

3 V. _  G5 U- l" vSo , in the optimal situation. ………

$ C0 G$ N, Y' W5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is
: o! m" q" P$ _* I3 E9 k7 w
3 x. O( H: B: M3 G. D+ \* g6 w公式

16． We use a similar process to find the position of the droplet, resulting in
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公式
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( |0 c+ N6 [; d2 hWith t=0.0001 s, error from the approximation is virtually zero.

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17．We calculated its trajectory(轨道) using

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1 _6 F6 B$ ?, \; D& V18．For that case, using the same expansion for e as above,
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5 N- g6 t- O9 {- E# f6 d+ N19．Solving for t and equating it to the earlier expression for t, we get
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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
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公式
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As v=…, this equation becomes singular (单数的).

 

9 J+ J' \! d% `* R2 h4 j1 N, l& O由语句得到公式
2 t' ]# Y; g7 K21．The revenue generated by the flight is
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+ q4 e# h; F+ b% }2 v% L. J公式
! v, q4 X9 q' m: k
 
6 J4 A6 i. U5 f; K& f' R. w4 W) Y
% S0 t7 G3 z! q: y8 p  N24．Then we have
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1 R( C" a' v& P公式
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We differentiate the ideal-gas state equation
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- V# ?3 v0 f% Z- s, b公式

Getting

9 F. m# e* v$ n' z- |8 o0 _公式

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

- ^+ H3 [- J9 o/ _  c7 B# z公式
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. @6 }0 X/ t; v
22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations
2 U0 E! p7 k/ K
公式

, ^8 |: F, k! I+ D  t  v2 L9 q9 oWhere P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）
) L$ v8 M8 s3 X2 g6 L/ I3 A
公式
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7 Z. U; r2 f7 r/ c% e版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
8 ]7 Z- O1 g. S# W; L% M9 V0 J: E原文链接：https://blog.csdn.net/u011692048/article/details/77474386
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