After running a few laps, Xu Chuan walked slowly to the pine forest canteen of P University.

It specializes in a variety of steamed buns and porridge, making it a great place to have breakfast.

After buying a few breakfast servings of fried buns, meat buns, soy milk, etc., Xu Chuan went back to the dormitory while eating, and at this time his teammates also got up, some were napping, and some had finished washing.

"Breakfast is here, everyone come and eat."

Xu Chuan put the fried buns and soy milk in his hand on the table.

"Thank you, Brother Chuan."

"Raw fried buns, love love, love."

"Brother Chuan has worked hard."

"What about fritters, Brother Chuan, there are no bad reviews for fritters!"

"Where's the bean juice? No bad reviews for specialty bean juice! ”

Hearing the breakfast, the dormitory instantly became lively, and everyone rushed over to divide the breakfast on the table.

Xu Chuan smiled, Songlin Canteen is bought with bean juice, but I'm afraid that no one else can eat that thing except for the locals.

And a bowl of bean juice is in the stomach, today's competition, people in this dormitory may not have to participate.

After breakfast, the group went to the competition venue.

The national decision of the competition is proposed and awarded by the National Competition Committee, and there are about 50 first prizes, 80 second prizes, and about 100 third prizes in each final.

In addition, there are individual special awards, such as the award for the best overall score, the award for the best theoretical score, the award for the best experimental score, and the award for the best performance of female students.

The number of students sounds like a lot, but in fact, it accounts for about half of the students who participated in the competition, just like this year, the total number of students participating in the national final is about 500.

However, the students who can participate in the national referendum can be said to be strong, and it is not a problem to award these students.

After the routine inspection, Xu Chuan entered the examination room.

The final round of the competition consists of two parts: theory and experiment. The duration of the competition is 3 hours each. The full score of the theory is 280 points, and the full score of the experiment is 120 points, and the organizing committee hires college teachers to score the papers.

The theory test is set at 8:30-11:30 a.m., and the lab test is set at 14:00-17:00 p.m.

Entering the examination room, the test paper was distributed, and Xu Chuan checked it first after getting the question.

There are a total of four theoretical questions for this year's national decision, one with 70 points, and there are at least three questions under each question.

It takes three hours to complete these four questions, which is quite difficult, and according to the usual finals, almost more than half of the students will not be able to complete these four questions in three hours.

However, this is used to screen talents, and it is normal to be a little more difficult.

After all, everyone can get a full score, so how can you tell who is stronger?

Therefore, there are very few people who get a perfect score in the national competition, and they may not be able to produce one in a year.

Xu Chuan didn't think much about it, and looked at the question after checking the test paper.

Question 1 (64 marks)

In June 2014, the CZ-2C launch vehicle was launched and successfully docked with the space station, which involved the encounter between the pursuer (CZ-2C launch vehicle) and the target (space station) in orbit around the earth.

In this problem, the Holman orbit change scheme is used to explore how the pursuer changes the velocity (velocity and direction) to achieve docking (encounter) with a target in a fixed orbit.

As shown in Figure 2a, both target A and pursuer c move counterclockwise at rate n in a circular orbit with radius ro, and their positions at time 0 are 0A, respectively. =0o，0i=0，rA; =rai=ro；

At this moment, the pursuer c ignites instantaneously, and the speed changes instantaneously △ (as shown in Figure 2b); The orbit of c is also from the radius r. The circular orbit instantly becomes the elliptical orbit shown in Figure 2c, and the major axis of the elliptical orbit is in the direction of the polar axis......

...... Objective: Aro Pursuer c Center Figure 2 aVo + Ava Pursuer cAv elliptical orbit circular orbit.

Question 1 (10 marks): If the mass m, energy E (which is actually the total mechanical energy of the projectile and the Earth's constituent system) and angular momentum L are all known quantities, try E, L, m and the known parameters To and 2o given in the question to represent the orbital parameters R and ε.

It is known that the form of the regular elliptical orbit (the long axis along the polar axis) under the polar seat R scale (the origin is taken as the right focus) is r(6)=1+ε cosφ, where R is the orbital size parameter and the orbital eccentricity, which are collectively referred to as the orbital parameter.

Question 2 (6 marks): Write the expression for the orbit Rc(0c) of the pursuer c after ignition (see Figure 2c), expressed by ro, eccentricity ε and φ.

Question 3 (6 marks): Write the ratio of the orbital period Tc of the pursuer c after ignition to the weekly TA of target A, Tc/Ta, expressed by ε and φ.

Question 4: (18 points) Define two ignition parameters (see Fig. 2b): the dimensionless velocity changes the angle between δ=|△υ/υ0|α, (when coincident α=0, the clockwise direction is taken as the positive direction), and the ignition parameters δ and α are used to represent the eccentricity of the orbit of the pursuer C ε and εcosφ.

Question 5 (9 marks): Consider the situation where pursuer C and target A meet at the confluence point of the first type of orbit (see Figure 2c). Let the number of times target A passes through the first type of orbital rendezvous from time 0 as nA, and the number of times that pursuer c passes through the first type of orbital rendezvous ......

Question 6 (3 points): Quantify nA with 8 and α, fix 8, and try to find two simple ....... for the change of function nα(α) relative to α

Question 7 (12 points): If you take one of the above two α0 values.

(1) There is an upper limit on the δ value, find ......

(2) Let the initial value of φA be .......

......

"It's still a familiar topic, a familiar formula."

The first big question came into his pupils, and Xu Chuan touched his chin with interest.

After the title came into view, those vague memories buried in the depths of his brain were trying to revive, bringing him a hint of familiarity.

Wujing is too far away from him before he was reborn, and the huge memories generated by twenty years are enough to bury these past events.

Even he doesn't have the ability to remember specific topics until now, but he still remembers the general range.

Although there are only four major questions in the national competition, the first big question will be transported to the town building first, and this cliff is to give the participating students a dismount.

One question, seven questions, no, it should be said that it is eight questions, and the difficulty soars directly to the sky.

Normally, there are often only four questions in the first question of the national decision, and the difficulty is relatively not very high.

But this year's first question, this is probably no less difficult than the finale of previous years.

"Do you want to get off the horse when you come up?"

Thinking about it, Xu Chuan looked up at the examination room.

As he expected, many of the students in the same examination room were a little stupid, looking at the test papers in a daze, and some of them frowned, and the signature pen in their hands unconsciously slid on the manuscript paper.

Those who can enter the national finals are basically students in the third year of high school, and they are all competition students, which means that all students have done the competition questions of previous years, whether they do it or not, but at least the topics and forms have been seen.

This came up directly to calculate the town building, and it was indeed a lot of competition students, so that many students didn't know how to answer for a while.

.......