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）
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公式
: H1 t) s. J5 x8 e: w5 ~" t
0 N" n! O) ?7 Z2 j9 SWhere

6 z1 U* n- A9 J3 c符号解释
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According to the assumptions, at every junction we have （由于假设）
" P# d6 }  x. v6 P, Y: Z. c
  C, k# S" g; I- @  H" j0 z6 `4 |公式

6 v8 ]2 G# a) N) R* e3 e由原因得到公式
& u6 p( s; x  z2．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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公式
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: `6 H  ]; D* z' h- J: @Since the fluid is incompressible(由于液体是不可压缩的), we have

: l6 O' {9 V. @! c公式
0 I5 {+ [1 M  m+ e( n1 [
1 D. R. C0 L' I7 C0 @, }Where
* z3 b. o+ N6 s9 ]: e& t
" L7 ~, {9 ^4 @% f. j0 A* w公式
  l/ L% r) q" C% {9 {  B5 V
用原来的公式推出公式
+ C* w4 g  b! D* b5 M' C& X# h5 Z3．Plugging v1 into the equation for v2 ,we obtain （将公式1代入公式2中得到）
0 e% n" M2 K* }  F9 h, y4 V
公式

. O, h7 r" m4 B* o11．Putting these together(把公式放在一起), because of the law of conservation of energy, yields：
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公式

12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）
( U6 v; }- g& ?" O: H7 C* ^
$ G2 b8 ?6 _& @% P& h) K$ Y公式

. N' O$ z% d2 RPutting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have

0 j* r6 J1 B6 f% S7 ]9 J: O' x公式
4 H  v$ h1 J2 @7 V' `) n9 w
) x8 P* l, O) O; M4 OPutting these into (1) ,we get（把这些公式代入1中）

公式

0 v9 j; y* E# o7 FWhich means that the
" q2 ], z8 W0 ?  \% `
3 C: J9 b1 b' hCommonly, h is about

+ [- Q5 X% R, f3 |From these equations, （从这个公式中我们知道）we know that ………
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) f9 Z1 s, f  p% B
引出约束条件
4．Using pressure and discharge data from Rain Bird 结果,
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+ X3 Y( ~1 s7 WWe find the attenuation factor （得到衰减因子，常数，系数） to be
8 u4 l* U% |- Z7 _5 c6 p/ y3 P
公式

$ l& R) n, m' I/ u( ~* D& r4 b计算结果
: z+ j0 |/ r4 n4 E! @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）
6 d. |! [+ D% X" W& L
公式
0 @3 k; B3 P$ f3 {9 F$ S, |4 c
Where
7 e9 w6 f! |1 M
; D# ~2 F9 W. k() is ;;
+ q9 ~2 K/ X4 k8 L. [4 W* f4 M
+ W# O* [; j+ |- d* }7．Solving for VN we obtain (公式的解)

公式

Where n is the …..

 

2 p/ w3 f% v$ Z- ]  G9 v8．We have the following differential equations for speeds in the x- and y- directions:
  P. l; m0 h' M, N; ~
) g* Z# @9 P( H5 k% v  u2 J* m公式

Whose solutions are （解）

公式

9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:
. n+ k" k& ?0 ~- }! a" M
, T; d, _1 \" ?3 X. f: r公式

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/ {, z$ R# I+ L3 h
公式
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3 b6 M, V/ u6 ]; d$ w# s! bThe decrease in potential energy is （势能的减少）
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5 h2 v* d" }5 R: M4 p. p公式

The increase in kinetic energy is (动能的增加)
1 @$ @. z, L) b( l; S
公式
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' x  y' E, i6 c0 ^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)
7 k. M3 Q, p2 {& G* l
Where a is the acceleration vector and m is mass
2 \  p1 X5 J) P& @3 {
  ?% y, f* J+ B9 C5 h 

" N% b# T  ~. f6 p- Z. ]Using the Newton's Second Law, we have that F/m=a and

/ |: V) k9 w/ y0 ^公式

So that
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4 c/ H. U! V6 Z7 O' R8 e公式

Setting the two expressions for t1/t2 equal and cross-multiplying gives
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4 q( r; v4 ?) h( Y- a/ G7 L公式
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4 u0 L: ~( \, n3 b3 _, R- Q/ r- `/ r) N22．We approximate the binomial distribution of contenders with a normal distribution:
5 [+ u( Z4 X% c. a
- U$ M' c& q8 A0 Q公式
0 M) \5 |) ]* [2 f! E  Y
Where x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives

7 _; `" R  F  l! p# D& |0 L1 ?公式

As an analytic approximation to . for k=1, we get B=c
* j: a, D3 g- \/ j) A' P, q
 

5 I# p5 t" A: }26．Integrating, （使结合）we get PVT=constant, where
- R: b! @5 _# \2 v+ J' ?
公式

4 k; o3 V$ Y8 {+ H* ~; e: yThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so

 
% M+ z+ o& Z+ U- t* A. `- S& s) y
- l, g' i2 O( P+ k23．According to First Law of Thermodynamics, we get
  e7 s9 ~' H8 r. ?, K) o
7 u/ S& h  @7 ^5 m8 N* V2 N公式

7 {2 d0 \+ i4 }& w0 ^# BWhere ( ) . we also then have
( f& X$ O. {) z! s" |$ c
+ r( r: G3 H7 |6 S3 T" \$ \1 T5 p8 j公式
: T1 ?: b  n7 B& w% T( l, b
& R. J( A$ f9 S9 m/ cWhere P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:

公式

Where

 

对公式变形
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到）
* K; y. c/ X. \3 \4 t
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

4 q- I0 o8 J- y+ Z3 @0 j1 C公式

! V- C6 a. A: @* f$ R0 q6 l4 w" N使服从约束条件
14．Subject to the constraint （使服从约束条件）

+ ?+ p) J6 C  [8 x公式
: f" y) L+ T4 m* T
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）
1 v0 m* e. C# V: h; [! \2 w
公式

- `5 N4 c! |- T4 {7 U5 z% J0 }5 GAnd thus f depends only on h , the function f is minimized at （求最小值）

/ K4 k% v! c6 B- S4 `. o公式

3 O; w. B: ]. w' ]At this value of h, the constraint reduces to
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公式
0 W7 H8 k; h  b, u) s. C3 Y
结果说明
# G, A, w0 D- _3 ]; ^$ }15．This implies（暗示） that the harmonic mean of l and w should be
1 B3 T6 c7 {3 J2 J- s
$ k* W. }. J. M# @, E公式
9 u# e9 ]/ B+ o+ n. B2 I0 J( I
5 [$ i9 L& H# W4 T9 NSo , in the optimal situation. ………
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5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is

公式
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! ^" K7 [: ~$ ~1 P16． We use a similar process to find the position of the droplet, resulting in
3 ?9 K/ g( [, g5 E- L8 @4 J
5 j  u; ^" n' K- H& w公式

With t=0.0001 s, error from the approximation is virtually zero.
% V6 f1 J5 l4 k4 t, A& Q! p
+ C3 j& k3 T' n. l7 n 
+ [, @6 Y' b0 v6 r
1 [6 V3 T; x( O: s& \4 F17．We calculated its trajectory(轨道) using

公式
8 i8 G# k9 ~0 F- o& b
4 }9 K5 T+ R' k' H: C* v0 G18．For that case, using the same expansion for e as above,
+ R1 X8 Q3 }6 G3 S( E- L- c5 H
公式
% e: r# r9 ^! l. b
19．Solving for t and equating it to the earlier expression for t, we get
0 f$ h3 v% p& e7 B
公式
2 [' q+ a0 V8 k1 B* q% a
6 V$ N8 L2 r' c; T20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is
/ L8 `( j* G7 q0 m
7 W8 T4 S& ]! X公式

& K6 B3 F' f( bAs v=…, this equation becomes singular (单数的).

 
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" H' e5 M% R4 b! z" O; Y9 V% l' P由语句得到公式
! j7 i% g* ]6 e8 K1 x21．The revenue generated by the flight is
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% i) k! n6 Z5 H5 n: h6 k公式

) {& g# |! |2 Y0 G 

& s! O/ y5 f$ h1 U' X24．Then we have

& J. Y" {; O9 w7 W. H$ X- O公式

We differentiate the ideal-gas state equation

公式
/ \6 P+ P+ W' Y* x; S
. [8 |0 C# u0 S6 k/ PGetting

. f* C1 u1 T, I: x( R公式

* C8 b1 t8 A! e  a& Q25．We eliminate dT from the last two equations to get （排除因素得到）
6 R, C) J- p4 p, Q- Y4 v
  L: q, _) m& H! h6 S% r' ~7 m2 O公式
, g1 O* w$ `5 w
& T" M5 Q& y. j! O6 P6 Z' x 
/ A9 l9 _5 v1 e1 r5 s
22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations
0 S& I& `& t+ l5 G( I
  f9 j( w& w( [7 ?$ |- Z8 f" h2 p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
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' Q) c0 _5 N. [, D$ I版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
0 U7 X. p3 i) d# E) L) v( g原文链接：https://blog.csdn.net/u011692048/article/details/77474386
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