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）

' ~+ }/ \  q3 @$ T$ _公式

+ \1 Y# n' f5 e& dWhere

符号解释
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According to the assumptions, at every junction we have （由于假设）

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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）；

公式
6 x! @( G. n& |6 P/ F- p) t: R
Since the fluid is incompressible(由于液体是不可压缩的), we have

3 E6 z! H9 E& [# P$ _! P1 J2 N公式
' R3 `# S$ a" v
Where

) Z3 |- e8 q* v- C' f: H( B/ W公式
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+ h  e2 G$ E- u1 K: A7 T( g1 h) ]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）
4 H3 q) u* \' M
3 ?/ r" S. S5 Y, U+ H公式

/ W4 X; \) `! u; q! P11．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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12．Therefore, from (2),(3),(5), we have the ith junction（由前几个公式得）
4 a# S7 X: P' d3 e) f+ h, y# C
3 k: w$ d( X5 S+ e公式

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

公式

# ~* D& |* T' t6 ^; _Putting these into (1) ,we get（把这些公式代入1中）

公式

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
5 c6 |) O9 ^" j: v4．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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1 k0 t/ I% I/ y. n- l+ y0 T& L# q公式
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# w! F% {) u/ k& r/ Z! K6 z# X计算结果
6 a/ ^: \. p# l9 a6．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）
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0 B1 z7 g! d5 d& k, @2 N- i& v7 Z4 x; e公式
* P+ w1 C" g* G) n1 N( B( b& w  d
Where
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() is ;;
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8 q0 a) o# N- Z7．Solving for VN we obtain (公式的解)

公式

$ D6 t- O4 |6 q8 k. n/ aWhere n is the …..
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5 u, V) `! h9 Q1 T# w  h+ F+ d# P5 r 
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8．We have the following differential equations for speeds in the x- and y- directions:

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Whose solutions are （解）

公式
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9．We use the following initial conditions ( 使用初值 ) to determine the drag constant:
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公式

! a6 I, C( [8 P6 l1 S( d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

公式

The decrease in potential energy is （势能的减少）
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公式
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The increase in kinetic energy is (动能的增加)
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+ f/ @5 O0 _$ w. i" ?& V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)

9 G' t2 F1 ^* g$ Y: _& eWhere 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

公式
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So that

公式

& ^( x) ?. Q  \" J7 Q) k! [Setting the two expressions for t1/t2 equal and cross-multiplying gives

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

& @8 F: F* i% o3 r' cWhere x is the cumulative distribution function of the standard normal distribution. Clearing denominators and solving the resulting quadratic in B gives
% G# T* i& N) U2 z: w& b
5 P. U, b! @' Z- ]2 P( k6 m6 p9 [公式

4 r5 P6 j! [% T9 f# ^6 OAs an analytic approximation to . for k=1, we get B=c
8 ]' V0 y1 M5 }7 t5 u6 t4 A
" j$ e" |; n% Q! k  R 

26．Integrating, （使结合）we get PVT=constant, where
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6 u6 ^* h/ N9 w% Y1 e  R公式
# d6 S, N5 h0 W4 G( n
# Y* U2 g& P: u" sThe main composition of the air is nitrogen and oxygen, so i=5 and r=1.4, so
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* O1 Z! o6 v% Q1 q" L) D23．According to First Law of Thermodynamics, we get

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Where ( ) . we also then have
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公式
8 ]4 y1 U% Z# P9 ^
) m  q* {3 E( V! D8 kWhere 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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公式
" b/ T* o- a' z) X' L5 _8 v. A$ N
6 f! W0 k: p) V" {" |; lWhere
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" r2 i4 t$ R: j) D( ?- C- p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

/ F" q* c7 _1 O+ n0 D/ W1 C$ v, ZWe 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

5 O0 L- g. q# y: y5 k% u9 `- q公式

4 K/ I# p9 V# D4 h# a; Z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）
4 u& ^! h7 \  O7 H' ]
/ C: t# {6 ?9 ^" D. R" T公式

! ?2 e; n* N2 ~9 v/ nAnd thus f depends only on h , the function f is minimized at （求最小值）
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公式
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At this value of h, the constraint reduces to

2 w: M8 p2 G9 K# z公式
: k" r+ [+ f$ C5 p% p# i. K
  n0 n6 j  Q* b- a7 s) k1 K结果说明
3 p% h& D. p2 B/ H$ |15．This implies（暗示） that the harmonic mean of l and w should be

公式

So , in the optimal situation. ………
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7 V, a$ O# H9 n3 S4 N5．This value shows very little loss due to friction.（结果说明） The escape speed with friction is
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/ _0 o  r) u6 a: E# w公式

  ]2 Z) W: K4 A" i) y6 r4 d16． We use a similar process to find the position of the droplet, resulting in

公式

9 B/ m1 H4 w8 F! Y# A: DWith t=0.0001 s, error from the approximation is virtually zero.
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2 ^+ O8 r, x2 @; i7 l17．We calculated its trajectory(轨道) using
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公式

18．For that case, using the same expansion for e as above,
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公式
( n+ T# t! h% I& J8 y9 L8 i2 V4 s
19．Solving for t and equating it to the earlier expression for t, we get
( w$ b1 x+ ^  G# s% Q) I( y
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/ R9 U$ L$ }. r+ m
20．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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4 v: S- N4 [: v9 Y( H; v, M由语句得到公式
* J: Q9 V0 @; N  x$ z21．The revenue generated by the flight is

7 p- q) S9 @0 z6 K# ]) c5 a: ^6 U公式
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/ z2 a$ D0 b4 |" W5 `1 r6 [ 
& G& l( D% {* |  e% S: D9 w4 }
24．Then we have
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公式
0 I0 o4 \3 h3 u, _$ t# N5 I
We differentiate the ideal-gas state equation
3 _& X/ [0 b+ x
公式

Getting
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9 K4 p5 S0 [3 n( x  M# F公式
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25．We eliminate dT from the last two equations to get （排除因素得到）

公式
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, r- j/ D" X* ^- X" ~+ v) e22．We fist examine the path that the motorcycle follows. Taking the air resistance into account, we get two differential equations
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$ O4 }' m- E1 W2 J* n4 `5 g公式
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* ?  w1 |& H7 O$ h; ~) k0 Q0 LWhere P is the relative pressure. We must first find the speed v1 of water at our source: （找初值）

3 B1 J* P8 P$ j$ B公式
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2 W: ?' u3 K  z8 K版权声明：本文为CSDN博主「闪闪亮亮」的原创文章。
2 N3 x% [3 T  @" l: W原文链接：https://blog.csdn.net/u011692048/article/details/77474386

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