美赛数模论文之公式写作

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由假设得到公式
2 r* T6 T/ u& G0 [3 {1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式）
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公式
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7 G2 a8 ]# f; p* g  b) _- S& d2 BWhere

符号解释
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2 I" j& d2 Q/ e9 p; uAccording to the assumptions, at every junction we have （由于假设）

. \6 w0 B  \( R9 m+ r- b; m) |5 ~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）；
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. G" E0 A6 G4 z; k  g" T! P, Y公式

+ G$ h. e# D% j1 \/ rSince the fluid is incompressible(由于液体是不可压缩的), we have
3 d% j" k6 [% y, S" ]4 F
$ m# ~3 ]% n1 B: s公式
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9 D# W8 m. q0 N9 h6 @' `9 nWhere
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公式

用原来的公式推出公式
$ R' u* d/ p6 F$ t2 c% k# k3．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：
. i0 J, S% c: Q( u$ K. V/ X" e* F9 \
8 {! _" F9 j2 k8 Z; \4 @公式

12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）

; ^' H) K5 b9 h; Q. m公式

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

公式
9 D, [: q* W/ Q5 C! X  K& f
Putting these into (1) ,we get（把这些公式代入1中）
& m( a! y1 H$ t
公式
! x/ R1 _& j% q- W$ Z/ r
Which means that the
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+ n' s5 C! x$ L0 m7 t, z# l& x4 KCommonly, h is about

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

 

引出约束条件
) P) \. a" C0 {' a3 u. O4．Using pressure and discharge data from Rain Bird 结果,
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4 `  b& {- X% z2 n9 j1 _* RWe find the attenuation factor （得到衰减因子，常数，系数） to be

公式

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）

公式

Where
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() is ;;

7．Solving for VN we obtain (公式的解)
: A9 [6 x5 b9 G: n( v
8 {1 c- {4 V9 l公式

* o7 r: J" T3 Q+ i1 R: S: n4 dWhere n is the …..
; E( [( e* r- ], S3 x0 \; ]1 Q
. o+ \! H  B9 M9 i- { 

8 c7 K% a7 x8 R, D8．We have the following differential equations for speeds in the x- and y- directions:

公式

- n+ s5 f* L( xWhose solutions are （解）

公式
1 |" `! V" `) n9 B- G/ P" U+ T
9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:
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公式
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: l! c# g3 ]$ _6 d' P根据原有公式
10．We apply the law of conservation of energy（根据能量守恒定律）. The work done by the forces is
  W+ n# g! q  A/ n
公式
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1 @7 r2 z0 _/ K2 o* w* g$ OThe decrease in potential energy is （势能的减少）
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公式
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* h# ]7 j7 ?! \' J, z: d  TThe increase in kinetic energy is (动能的增加)

公式

Drug acts directly against velocity, so the acceleration vector from drag can be found Newton's law F=ma as : (牛顿第二定律)
1 b* A" b# k8 U9 @
$ R. o) K4 u. F+ J& _& i! }Where a is the acceleration vector and m is mass
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Using the Newton's Second Law, we have that F/m=a and
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公式
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So that

公式

% l' a) ]* I" @) r' H% R0 n) M3 T2 f; vSetting the two expressions for t1/t2 equal and cross-multiplying gives

公式
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/ L( s7 L9 H) z" N+ g9 L* r22．We approximate the binomial distribution of contenders with a normal distribution:

# \1 g* G# Y# z3 e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

 

- D4 S4 ?0 d: g2 @- F26．Integrating, （使结合）we get PVT=constant, where
- ^* \9 T$ r7 m/ D5 N% Y5 N
公式

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

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* L; D! u- `9 V) R, D23．According to First Law of Thermodynamics, we get
$ r  B+ s: w0 w0 @( G4 }& C
公式

Where ( ) . we also then have

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Where P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:
/ p( H" l7 M1 C5 a
4 F3 j5 d, k& m! G  x! F+ X公式

Where

. ~5 @* x( _4 k* Q+ Q 
+ I4 ^8 F/ }* d2 k; u
; ?3 ]( ]- z6 e# K& g% q对公式变形
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到）
+ a. N9 L! U; j
0 a. T! r& t3 x6 S6 E" S# y7 m公式

, q# A7 R$ p, BWe 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
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, N% D, H# U. {公式

' {/ e% I9 Z0 A  ]  c5 m使服从约束条件
14．Subject to the constraint （使服从约束条件）
, l- p+ A' R# }8 B
; H* b, i# D( S, m) c公式
, z0 l& B+ Q6 K* k
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）
6 ]) X8 j. m* l' S7 H% {( \
* \0 v1 H6 k( @+ o7 B7 \0 f公式
1 p( T: q5 I7 y5 c0 \
) U+ O* j9 K5 @% p1 @( _And thus f depends only on h , the function f is minimized at （求最小值）
+ D) a: w  n$ x
, _+ \  K8 w* z" D) t8 R公式
) e+ n( C1 _$ ]5 L5 i; u" c. ~( x
( E5 d7 I7 S7 O7 K: gAt this value of h, the constraint reduces to
3 i0 d/ X, R/ d, C6 U6 v
% w  r+ b4 `* L公式
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结果说明
15．This implies（暗示） that the harmonic mean of l and w should be

6 G, b% _" q4 k( C公式

1 r- D9 l6 i: b) A) z# y' I3 c# YSo , in the optimal situation. ………

5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is
7 k+ j5 C- j& q; w. f1 y
公式
! ~+ V: {8 q* c" J3 N
16． We use a similar process to find the position of the droplet, resulting in

公式

/ ^' K# R% i* I2 N9 u# h4 ?! f4 _With t=0.0001 s, error from the approximation is virtually zero.

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8 z7 n% V* ~8 x17．We calculated its trajectory(轨道) using

+ ]7 r& _) [6 Q5 j+ j& O公式

/ u. w( G3 E' S1 K18．For that case, using the same expansion for e as above,
- G2 c5 H% w) E
( p9 l6 k' Z/ R9 _公式
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19．Solving for t and equating it to the earlier expression for t, we get
6 b/ `) h& X$ c2 c  G6 Y
8 U7 }7 W3 ]. L! x  h7 l7 I公式
7 c% G3 I7 b. F2 H$ c
% s) q2 U+ N: R$ b6 X20．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
- b- A* F8 L+ |% W' s
As v=…, this equation becomes singular (单数的).

 
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由语句得到公式
* ?: U; ?" _9 `21．The revenue generated by the flight is

公式
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6 L( k+ w& D2 i$ M0 I! v5 m 
: \9 E8 j2 {$ N$ P8 `' W
* z2 }/ ~7 L; _  {9 V; D% `# {' A24．Then we have
0 h# F4 l9 v+ h
公式
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* `& O' a5 }1 ^: X( LWe differentiate the ideal-gas state equation

+ Z) d1 Y% |% [* t  B公式
9 h0 i1 }  r2 k; @& ]* \; {; S: I
. R9 b3 Y+ c7 [" c8 t2 F0 \& ^Getting

公式

25．We eliminate dT from the last two equations to get （排除因素得到）
+ G* q0 ^9 Q  D6 v
9 x4 U2 T0 m% w5 n0 f! w公式
; e# L- A- q5 J' m6 y# N+ p3 i
 

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

+ t% F0 a  }* i% p: n5 N; A7 y' u公式

: B8 A8 k, d* HWhere P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）
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公式
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" j, h  Z, g$ `+ J! H. M版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
5 J) f5 P' Z# ^, \8 A! L" E原文链接：https://blog.csdn.net/u011692048/article/details/77474386

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