美赛数模论文之公式写作

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由假设得到公式
% b" ]3 e( A7 @$ K* @, W1．We assume laminar flow and use Bernoulli's equation:（由假设得到的公式）

公式

3 _( ]; C" D" x3 L' @Where

符号解释
  b* x6 D; u& H# z& X( K
According to the assumptions, at every junction we have （由于假设）

公式
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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）；

; G# C, [+ j2 _7 b公式
+ y+ E/ O! ^5 K" U. z) W' ^0 O  {7 a$ I
% e9 [. I7 B! [- n* v( L1 d/ tSince the fluid is incompressible(由于液体是不可压缩的), we have
6 |  |. c* S! A3 o7 H1 F
公式

+ _  J, g- }  N3 B4 WWhere
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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）

公式
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11．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
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12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）
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- L+ \9 N' F6 f+ F! x7 K/ c公式
- ]" O, E! ~: G$ Z/ I+ @2 i
) r: r3 O# B$ W) o& f0 QPutting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have

( x' d0 A* G: c9 ^6 f7 D公式
8 t. O4 r6 U; z, m3 N$ Z+ E) U3 M
Putting these into (1) ,we get（把这些公式代入1中）
# k% ^0 G" I( Q4 R: _/ n2 V7 Q3 @
公式
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+ G" F+ l; |. k2 q5 }! {1 M& MWhich means that the

Commonly, h is about

+ f4 m  x4 [6 _7 [3 V! d1 L/ `From these equations, （从这个公式中我们知道）we know that ………

 

% e7 U4 o. J! k9 K- ~引出约束条件
/ `- D- W6 N. d4．Using pressure and discharge data from Rain Bird 结果,
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% O3 u3 c1 a$ i6 U# v$ H4 W% Y' bWe find the attenuation factor （得到衰减因子，常数，系数） to be
5 Y+ Z* Z' @1 Z$ C+ s+ V. ?* z
公式
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计算结果
" b( y% J% o. ]8 \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）

: y, f$ j+ j& B. _0 g2 J公式

4 d1 Z4 R4 {  V' I, dWhere

() is ;;

: h3 h9 K/ w( K* N, i7 d" I/ F7．Solving for VN we obtain (公式的解)

& H+ W0 C6 A. Q& e# s公式
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Where n is the …..

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

: p  I$ d( h) J& sWhose solutions are （解）

1 t6 i  `, K: ]公式
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9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:

公式
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2 f: F: F* u0 X2 m  Q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
. d" i2 r, w% z4 i
3 B  I$ L! s7 }; a, p- q公式

" L" F, f$ [! {7 PThe decrease in potential energy is （势能的减少）

公式
2 ?5 y& B$ n9 c/ F/ {7 R& p
. Z2 y5 W! ^3 `- m) h8 CThe increase in kinetic energy is (动能的增加)
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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)
' c- o. o% j; s- G- n; ^# q
- Q: J5 x: d2 M9 F1 J$ {Where a is the acceleration vector and m is mass
8 |. u3 a+ V% s' Z2 T# q# v  _
 
# j, ^6 o8 w: h6 e! y- L$ R
# y0 P8 k, X+ @7 u, D+ NUsing the Newton's Second Law, we have that F/m=a and

公式
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So that

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

公式
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22．We approximate the binomial distribution of contenders with a normal distribution:
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4 F* Y  [3 g3 X! h公式

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

4 R3 n1 {6 z+ C* i, K* @公式

: C0 F% F" M4 d  P( T/ JAs an analytic approximation to . for k=1, we get B=c

 
+ y! h4 L! v, r8 u
3 v; z* o0 V6 F( M" D" _1 C26．Integrating, （使结合）we get PVT=constant, where

5 F; E: b' m: R2 c5 J3 Y公式
2 E% {+ P1 G/ X' [
The main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so

 

2 O$ @) Q( c8 k0 r# ^9 x( P23．According to First Law of Thermodynamics, we get
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公式

" h9 `; f8 j' e* \0 Y( _Where ( ) . we also then have

公式

" W: L( X. I/ O0 m/ s6 Q3 }* \, U7 {% XWhere P is the pressure of the gas and V is the volume. We put them into the Ideal Gas Internal Formula:
) |! d2 v/ A: ]# K0 Y9 l% ~
公式

Where

9 b, p" s$ Q) c2 X! J* x. u/ I 
' B  @9 Y3 T% s* L
对公式变形
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到）
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公式
9 a5 ~2 r0 k* O7 d( U
5 g* n: E3 E4 z) t0 T) q2 U! s3 nWe 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

  a3 u: u; B1 C公式

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

  W6 a8 s) }6 y) _- t. `. d6 N  b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）

  u3 y7 j$ {% T- @* M公式
* t$ g, l- t5 p( ^8 [
$ J  U0 ~0 X+ k3 QAnd thus f depends only on h , the function f is minimized at （求最小值）
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公式
4 D8 h/ i) L, Z, @2 @; h
" [8 H5 O/ O' f. y% Z  b) P$ F/ lAt this value of h, the constraint reduces to
8 O/ h4 x7 D! s! o
公式

. h( x2 u5 Y; a, o结果说明
15．This implies（暗示） that the harmonic mean of l and w should be

0 d. e% M0 E0 f9 L公式

So , in the optimal situation. ………

5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is

6 V! b+ G4 J* d5 }! p" K8 Z公式
0 T+ b% x7 ^2 K2 A1 ~9 F$ U
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.

 
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17．We calculated its trajectory(轨道) using
% H- z9 j% k  \) n( Y  X
公式
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- C. X9 D' ?! l& {) x18．For that case, using the same expansion for e as above,

公式
; R% v/ D0 w# e5 h/ B5 s+ ?! V
19．Solving for t and equating it to the earlier expression for t, we get

公式

" n6 D+ M  h) m" ?/ s" _20．Recalling that in this equality only n is a function of f, we substitute for n and solve for f. the result is
' @2 S8 V0 G- W8 a- G8 d9 P6 s) Y
- u: t1 O6 t; x& ]9 K% U公式

As v=…, this equation becomes singular (单数的).
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由语句得到公式
21．The revenue generated by the flight is
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公式
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$ o* a/ V, V/ h2 R1 d! ]0 N, R 
) Z8 e. ]9 d% ]$ U" K- f4 ?
2 ]; }4 m2 U1 E24．Then we have
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3 {' i: P7 A$ T) [* X' f1 ^公式

  [: ]/ r) a0 gWe differentiate the ideal-gas state equation
) M3 r1 X6 b$ U) p$ ]7 _
公式

Getting

0 j+ G  x! D/ ^. S+ f8 P公式
( S4 c( H" i2 G' C
( n3 i9 V5 l  Z4 T25．We eliminate dT from the last two equations to get （排除因素得到）

1 J+ U$ d; q! ~$ Y& c5 l公式

 
7 h2 v# b- r, _4 m" K
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
% ?0 X* C* {/ t' b% b1 \6 {————————————————
3 U. g6 i9 n9 y& R% `' n版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
& c! v. `1 q2 p9 W# e* L原文链接：https://blog.csdn.net/u011692048/article/details/77474386

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