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）

3 S/ a1 @7 E; n8 G, _公式

Where
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符号解释
8 `: R0 @% [/ M& a( ]7 k( z% U
, r( W& n+ J/ o( eAccording to the assumptions, at every junction we have （由于假设）

" {3 n; a7 |0 A% c! j公式

/ ~7 i# R6 f( U8 @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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公式

  s! u& q! Y5 E, V3 i$ TSince the fluid is incompressible(由于液体是不可压缩的), we have

3 i6 H% t- O% c' B" b公式
: L( a) m4 z. H$ N& Z  P
. }" A4 Y5 `5 s) `/ zWhere
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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）

6 ~  f* W; @1 o1 \1 p+ a0 m公式

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

公式
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* J4 R) ?# ?3 D# E2 _  P, ~12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）
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公式
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4 w2 d2 H. c5 g% p6 E& p- WPutting (1)-(5) together, we can obtain pup at every junction . in fact, at the last junction, we have

公式
; q3 O+ R2 d# l2 I* ^. t3 j( f* l
- i: ~0 `9 K9 O9 \; R9 l  [Putting these into (1) ,we get（把这些公式代入1中）

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Which means that the

Commonly, h is about
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From these equations, （从这个公式中我们知道）we know that ………
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引出约束条件
4．Using pressure and discharge data from Rain Bird 结果,

9 C* R; [) ?3 w( WWe find the attenuation factor （得到衰减因子，常数，系数） to be

' B0 E/ c, T3 }2 G3 a5 j' e- p公式

0 `" L7 }" _6 G$ a1 T# s/ A) n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）

" C2 V" n7 J5 y7 U! U! j公式
2 M' d" i1 `7 a2 b& c1 x
& S$ Y" \7 N% Q0 h0 ZWhere
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+ \( X) t2 U" c0 j# [8 ]0 Q() is ;;

7．Solving for VN we obtain (公式的解)
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公式

" g% M6 A7 l6 x# y# @, E) {& qWhere n is the …..

 

8．We have the following differential equations for speeds in the x- and y- directions:
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6 _  z8 P! N4 I; [5 S公式
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) R" d& I) A& `; o0 @3 g( B3 W; IWhose solutions are （解）

公式
+ l# ?( a# ~& L% F
3 R3 }2 F$ w. }  Y6 B9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:

公式

6 [2 b9 D$ {7 x$ `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
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公式
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  w5 p! ~: u4 d5 SThe decrease in potential energy is （势能的减少）
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公式
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0 j8 G$ _- n9 Q+ IThe increase in kinetic energy is (动能的增加)
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公式
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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)
: ~9 B) L# G" u+ j9 F+ Q
Where a is the acceleration vector and m is mass
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' o: \/ u  A6 F  w
Using the Newton's Second Law, we have that F/m=a and
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$ B8 M- Z9 i. e公式
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So that

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& H+ Q: ]( Y; m& T8 _5 DSetting the two expressions for t1/t2 equal and cross-multiplying gives

公式

22．We approximate the binomial distribution of contenders with a normal distribution:

公式
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* H8 c! T" {7 z$ yWhere x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives
2 X/ B4 X( v" }! [, S. X5 r
( @5 b8 B3 Y# A" T" `+ P3 u6 P公式

As an analytic approximation to . for k=1, we get B=c

 

26．Integrating, （使结合）we get PVT=constant, where

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6 @7 `" S& w. l& ^7 T( m( i7 wThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so

 
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. w1 T9 p9 J  ^, f0 O% A- I! ]+ @: M9 ?# U23．According to First Law of Thermodynamics, we get

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Where ( ) . we also then have
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( a* a# @1 [+ i9 @& B公式
0 n. t; M. {8 B$ Q
Where 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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" g2 O  _, n5 k% g: j公式
2 {  b6 g/ R6 l0 P1 I' V
9 B  @0 c7 g# H% z2 GWhere

 

6 j# v* h* |% w5 [8 ]对公式变形
13．Define A=nlw to be the ( )（定义）; rearranging (1) produces （将公式变形得到）
6 M) X* f0 j/ q6 v, z) C
公式

$ a  _2 j! I8 mWe 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

/ P% A' K, p6 I' t0 B2 \0 @1 U% M公式
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4 |* h) p$ e. ]. o% F. V使服从约束条件
14．Subject to the constraint （使服从约束条件）

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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）

公式
: x1 \# r! N5 b, x( G) V- L
/ @+ v+ J8 W% b/ D1 v* q; RAnd thus f depends only on h , the function f is minimized at （求最小值）
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# j0 x( u1 B% Z2 A公式

At this value of h, the constraint reduces to

公式
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结果说明
1 O/ z9 u1 M; U) A+ j$ e/ V) T0 N* g15．This implies（暗示） that the harmonic mean of l and w should be

公式

( s  d( \! a  y1 F, S  e9 DSo , in the optimal situation. ………

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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16． We use a similar process to find the position of the droplet, resulting in
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# Q1 D# R% s  _9 T公式

- K$ G) F* @8 s: h: F# vWith t=0.0001 s, error from the approximation is virtually zero.
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9 J3 e; M) \# N5 X) c 

17．We calculated its trajectory(轨道) using
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公式
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18．For that case, using the same expansion for e as above,
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公式
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19．Solving for t and equating it to the earlier expression for t, we get

( X& t; W% w" C. f+ B公式

( w) ^; Y% z: v20．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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3 S( Y/ j9 e4 h* U) y公式

As v=…, this equation becomes singular (单数的).

 

% W, w3 ?; p3 t! Z" r' I8 F8 r由语句得到公式
% Z: @; a  S$ z21．The revenue generated by the flight is

公式

 

24．Then we have
( I( L+ s' l, L: _
# F/ E& x& s, _) t4 z2 e公式
" I- V: r5 s, U+ ^
* p; O6 [* J% IWe differentiate the ideal-gas state equation
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& |" w6 a& Y0 R% b2 A公式
1 Z8 U& d. t1 w7 l) V4 P' L
Getting

' f6 Q9 B- p+ p; G- k4 N公式
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" d! L+ ?0 j* G, P8 p$ a7 x25．We eliminate dT from the last two equations to get （排除因素得到）
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公式
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/ Y* N. y  [0 }7 _% K3 }22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations
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公式

Where P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）
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0 \; Z% t" ?# [+ v5 X* Z) X公式
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版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
' y% v2 F5 F0 q原文链接：https://blog.csdn.net/u011692048/article/details/77474386

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