美赛数模论文之公式写作

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由假设得到公式
6 C4 t) y/ v$ d% Q( E1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式）
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公式

8 ?1 E( \; K+ R1 Q- x6 oWhere

3 \7 `% b. H# C; k5 S9 Y$ g符号解释
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According to the assumptions, at every junction we have （由于假设）

5 X# M; o. {3 c5 e% f: O公式
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由原因得到公式
. h3 j* _4 `3 J$ V. V( `  p- H  d2．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）；
1 O( o# q& K9 v6 A* i$ H9 b
公式

Since the fluid is incompressible(由于液体是不可压缩的), we have
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公式

Where

公式
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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）

3 |! Z1 R. V/ _; [/ p6 W$ v公式

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

3 ]) c4 v9 S( x9 w8 x公式

' {) T# ]$ K# k4 ^+ i) \12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）

公式
+ v1 \& k% y' }* K  `6 M
Putting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have

  j3 u- q3 x+ Y/ V5 O# q# f# u公式
- u3 w* c& ~% {/ K) v
Putting these into (1) ,we get（把这些公式代入1中）
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公式
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) I* t% w6 \3 d6 \* S4 b' ^! h8 WWhich means that the

8 j- `5 v6 a6 h+ \4 YCommonly, h is about

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

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: t$ R5 I5 ]9 |/ z: e  _6 w' K引出约束条件
4．Using pressure and discharge data from Rain Bird 结果,

We find the attenuation factor （得到衰减因子，常数，系数） to be
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, t+ c2 _% X) ?, j1 Y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）

& u' b  O8 S9 [; b2 X& v公式

: }' w) N6 l/ L% T! HWhere
4 j: I1 e3 k1 f  w, b# s7 ~+ h) y
5 V/ r8 {& ?  P0 E() is ;;
* G7 T5 }& g- J8 a+ d2 |
, ]! Y# Q  E) H* N2 i( a7．Solving for VN we obtain (公式的解)

6 |% a% z) \6 R) q7 `7 p公式

Where n is the …..

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

& @) e, F4 Z' O) D公式
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6 b$ }. v1 |+ }! E( _: h/ ^Whose solutions are （解）
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+ a! F7 x. y$ t( o- Y. k公式
7 ^1 Q* m- W( r% }' ?5 h
6 W0 [2 I8 r) Z2 h' \6 e9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:
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! `. O7 u; J# a1 d+ p8 d. c& e公式

+ ?4 k. t: z6 R" y& G: \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
3 y  M; `! \+ {5 B
公式

6 {2 o/ u. T5 S9 |: XThe decrease in potential energy is （势能的减少）

公式

  e5 S0 T. k1 B5 XThe 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)

" O& b, l% L% r. g" \" J* rWhere a is the acceleration vector and m is mass

 

! T0 s- k6 d6 O7 g: M# v; aUsing the Newton's Second Law, we have that F/m=a and
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' D4 d) ~1 O4 C! p; N* U0 Y5 q5 D公式

So that

5 [9 q% E  T: x0 ~) u2 x3 B公式
  \& F) h" b- r* o1 P7 [, H% A0 a
7 d% b& ~& b$ H0 A2 u, ~/ mSetting the two expressions for t1/t2 equal and cross-multiplying gives

公式

3 v( v3 y4 R8 V* ~$ i22．We approximate the binomial distribution of contenders with a normal distribution:

6 j6 k' P. j  X! ^: z公式

: C6 {- |1 [7 G( X% y4 DWhere x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives

4 ~) M) e" O/ r0 V$ Z# K% L公式
8 X1 o' y# t4 L  R5 {1 Q* s
- F6 v7 N$ b4 `As an analytic approximation to . for k=1, we get B=c
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( H3 y( A" f! w1 p# h# l
: [4 N7 R& G: J! k7 M26．Integrating, （使结合）we get PVT=constant, where

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

 

! T' |) n! l- c$ @+ p* n23．According to First Law of Thermodynamics, we get

公式
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- P/ {5 P& j  b. {Where ( ) . we also then have
+ Z5 a7 R9 E6 Z% ]: W; w+ i# Y
  O! L9 B# I- k2 o7 \0 s公式
* w, ^2 U/ ]0 t2 K  A9 C
) |! O% D4 K. i9 z. JWhere P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:
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' }$ k8 {0 r1 b# }' D$ B公式
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Where
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  {3 m+ s8 B+ l6 K) s8 ?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' ^9 U5 m: f公式
( u  F- G* }" R/ n
1 g8 @; k) f4 J. V% 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

公式
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使服从约束条件
14．Subject to the constraint （使服从约束条件）

公式

  L  u" ^( P! V: a, c0 S) AWhere 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）

# v4 Q% `1 q( ]3 b( t9 n2 Y7 p+ J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
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" P- @7 w! N7 X% z5 @公式

2 h& X* s! u9 V. c9 V结果说明
& A, M% F" p, y. E15．This implies（暗示） that the harmonic mean of l and w should be
9 @5 n- [) W" i0 X
3 Q! n& A% w/ L' M公式

- p0 g, d4 h8 ?5 l0 B- v9 _3 ^So , in the optimal situation. ………

3 a- J6 W2 s/ A8 E5 T% y8 {2 @7 D5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is

4 s  S* H- t! t* {; f( Z公式

2 ~9 K- m$ w: m16． We use a similar process to find the position of the droplet, resulting in

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With t=0.0001 s, error from the approximation is virtually zero.
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17．We calculated its trajectory(轨道) using

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

8 p8 ]3 f* Q% K! d公式

9 |# s+ F2 q3 l19．Solving for t and equating it to the earlier expression for t, we get

公式

: A& B) L' ^0 Y, ^7 B: z20．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 (单数的).
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5 x9 E) `, ]% g& H, `7 ?/ ]1 j, w! M由语句得到公式
21．The revenue generated by the flight is

公式

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& I9 G* E6 {- k) a. Q, Y7 I
  G% f  I6 J' G0 N6 X. v24．Then we have
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公式

, j; H. \' F. Z1 Y! e, Q) rWe differentiate the ideal-gas state equation

公式
9 d* o: G( H6 x! o' n5 A
Getting
0 n+ H2 I1 g0 ]+ q2 x/ X  \3 a2 a
8 I) M, K3 L) y5 p* S. \公式

8 S# Y! m% }4 ~% y# M1 T, p4 W& Q! o25．We eliminate dT from the last two equations to get （排除因素得到）

公式

 
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22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations
5 m1 T, ]) X- E+ N5 V2 K
公式

3 f0 e; l' W, F. ^Where P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）

5 j2 e7 {# n, ?3 [: x4 r* X- [, @公式
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0 ?- {5 T1 u; e2 V% }版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
: }  J6 l. x" I& H原文链接：https://blog.csdn.net/u011692048/article/details/77474386

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