Great Country Academician

After running a few laps, Xu Chuan walked slowly to the Pine Forest cafeteria of P University.

The main business here is a variety of steamed stuffed buns and porridge, which is a good place for breakfast.

After buying a few fried buns, meat buns, soy milk and other breakfasts, Xu Chuan went back to the dormitory while eating. At this time, his teammates also woke up one after another. Some were waking up and napping, and some had finished washing.

"Breakfast is here, everyone come over to eat."

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

"Thank you Chuan."

"Sheng Jian Bun, I love it."

"Brother Chuan has worked hard."

"What about the fried dough sticks, Brother Chuan, there are no bad reviews for fried dough sticks!"

"Where's the bean juice? There's no bad review for the specialty bean juice!"

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

Xu Chuan smiled. Pine Forest Canteen sells soybean juice, but I am afraid that except for the locals, no one else can eat it.

And with a bowl of soy juice, no one in this dormitory may have to participate in today's competition.

After breakfast, the group headed to the competition venue in a mighty manner.

The national finals of Wujing are determined and awarded by the National Competition Committee. For each final, there are about 50 first prizes, about 80 second prizes, and about 100 third prizes.

In addition, there are also individual special awards such as the Best Overall Achievement Award, the Best Theory Achievement Award, the Best Experiment Achievement Award, and the Best Achievement Award for female students.

The number of people sounds a bit high, but it actually accounts for about half of the participating students. Just like this year, there are a total of about 500 students participating in the national finals.

However, the students who can participate in the national finals 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 of Wujing consists of two parts, theoretical and experimental. The competition time is 3 hours each. The theoretical full score is 280 points, and the experimental full score is 120 points. The organizing committee will hire university teachers to mark the test papers.

The theoretical examination time is set at 8:30-11:30 in the morning, and the experimental examination is set at 14:00-17:00 in the afternoon.

After entering the examination room, the test papers were handed out. After Xu Chuan got the questions, he checked them first.

There are four theoretical questions in this year's National Finals, each worth 70 points, and each question has at least three small questions.

It took three hours to answer these four questions, which was quite difficult. According to the usual finals, almost half of the students could not finish these four questions within three hours.

But this is used to screen talents, and it is normal that it is difficult.

After all, everyone can get full marks in the test, so how to distinguish who is stronger.

Therefore, there are very few people who get full marks in the Wu Jing Guo Jue, and there may not be one in a year.

Xu Chuan didn't think too much, and looked at the questions after checking the test paper.

Question 1 (64 points)

In June 2014, the "CZ No. 2 C" launch vehicle was launched and successfully docked with the space station. This involves the encounter between the pursuer ("CZ No. 2 C" launch vehicle) and the target (the space station) in orbit around the earth.

This question uses the Holman orbit change scheme to explore how the pursuer changes the speed (speed and direction) to achieve docking (meeting) with the target on the fixed track.

As shown in Figure 2a, the target A and the pursuer c are both moving counterclockwise at a speed n on a circular orbit with a radius of ro, and the positions of the two at time 0 are 0A; =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 also changes from a radius of r. The circular orbit of the ellipse instantly changes into an elliptical orbit as shown in Figure 2c, and the long axis of the elliptical orbit is in the direction of the polar axis.

Target Aro Chaser c Center Figure 2aVo+Ava Chaser cAv Elliptical orbit Circular orbit.

The first question (10 points): If the mass m, energy E (actually the total mechanical energy of the system composed of the flying object and the earth) and angular momentum L of the flying object are known, try E, L, m and the given amount Known parameters To, 2o, etc. to represent orbital parameters R, ε.

It is known that the form of a positive elliptical orbit (the long axis is along the polar axis) under the polar coordinate R (the origin is taken as the right focus) is r(6)=1+ε cosφ, where R is the orbit size parameter and is the orbit Eccentricity, collectively referred to as orbital parameters.

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

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

Question 4: (18 points) Define two ignition parameters (see Figure 2b): dimensionless speed change δ=|△υ/υ0| angle α between them, (α=0 when coincidence, clockwise direction Take it as the positive direction), use the ignition parameters δ and α to represent the eccentricity ε and εcosφ of the track of the pursuer c.

Fifth question (9 points): Consider the situation where the pursuer c and the target A meet at the first type of orbital meeting point (see Figure 2c). Let the number of times that the target A passes through the first type of orbital meeting point since time 0 be nA, The number of times that the pursuer c passes through the meeting point of the first type of orbit.

Question 6 (3 points): Express nA as 8 and α, fix 8, and try to find two simple ways of changing the function nα(α) relative to α.

Seventh question (12 points): If one of the above two α0 values ​​is chosen.

(1) There is an upper limit to the value of δ, 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 great interest.

After the topic came into view, those vague memories buried deep in his brain were trying to recover, which brought him a sense of familiarity.

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

Even he is not capable of remembering the specific topics until now, but he still remembers the general scope.

Although there are only four big questions in the Wujing Guojue, the first big question is to move to the town building first, which is definitely a blow to the participating students.

One question, seven small questions, no, it should be said to be eight small questions, the difficulty is directly skyrocketing.

Normally, the first question of Wujing Guojue usually only has four small questions, and the difficulty is relatively not very high.

But this year's first question is probably no less difficult than the last question in previous years.

"Are you going to disown Mawei as soon as you come up?"

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

As he expected, many students in the same examination room were a little dumbfounded, staring at the test paper in a daze, and some of them frowned tightly, sliding the signature pen on the manuscript paper unconsciously.

Those who can make it to the national finals are basically high school seniors, and they are also competition students. This means that all students have done the competition questions of previous years. Regardless of whether they can do it or not, at least they have seen the topics and forms.

This directly moves the track to calculate the town building. It really stuns many contestants, and many students don't know how to answer it for a while.