pjbooks Chapter 26: A Perfect Score Kill!
Chapter 26: A Perfect Score Kill!
In the office, Fang Jianyu had just taken a sip of water from his enamel mug when he heard Wei Jiqiu's 20-minute time limit and almost choked.
twenty minutes?
He glanced at the problem on the blackboard; it was a competition problem with two sub-questions and a function graph.
Even the top students among those competition participants that Old Wei himself mentors probably don't have enough time, right?
He coughed, instinctively wanting to say something, but seeing Wei Jiqiu's expressionless face, he swallowed his words.
Old Wei never negotiates the price of his exam questions, and the top students receive the same treatment. He can't make an exception just because Han Chuan is "one of his own."
However, Han Chuan didn't care. He reached out and took the chalk from Wei Jiqiu, and stood in front of the blackboard.
A small ball of mass m is released from rest from the top of a smooth quarter-circular arc track of radius R. The end of the track is tangent to a rough horizontal surface of length L with a coefficient of kinetic friction μ. The other end of the horizontal surface is connected to a smooth inclined plane with an adjustable angle θ. The maximum height the ball rises after sliding up the inclined plane is h.
Given m = 0.5 kg, R = 1 m, L = 2 m, μ = 0.2, g = 10 m/s², and neglect air resistance.
[Question 1: If θ = 30°, find h.]
[Question 2: To ensure the ball returns exactly to the midpoint of the horizontal plane, find the range of values for θ.]
[Question 3: The inclination angle θ of the inclined plane is continuously adjustable from 0° to 90°. Find the functional expression h(θ) of h with respect to θ, and draw a rough graph, marking the key points.]
A standard competition question, consisting of three sub-questions.
Although he couldn't determine whether it was the difficulty of the preliminary or semi-final rounds, based on the competition rules that Ge Jun had taught him before, this shouldn't be the final question; it was more likely the first or second problem in the problem-solving section.
Because the final question in a competition usually has five, six, seven, or eight sub-questions, three sub-questions are obviously too few.
Instead of rushing to start, Han Chuan first stared at the questions on the blackboard.
Wei Jiqiu and Fang Jianyu watched quietly from the side.
Time passed by, little by little.
Just as Wei Jiqiu frowned, wondering if Han Chuan was having trouble coming up with a solution,
Han Chuan finally moved in front of the blackboard. He picked up the chalk and quickly began to write on the blank area of the small blackboard.
【untie:
For the first question, Ek·sin²θ = mgh, which means 8sin²θ = 5h. When θ = 30°, sinθ = 0.5, so h = 0.4m.
[Second question: μm·(L/2) = Ek_return, Ek_return = 0.5×10×0.2×1 = 1J.]
[Ek·sin²θ= Ek_return = 1J, that is, sinθ≥√(1/8)≈ 0.354, θ≥ 20.7°. 】
The range of θ is 20.7° ≤ θ ≤ 45°.
The answers to the two sub-questions appeared quietly on the blackboard in less than three minutes. Faced with the most difficult third question, Han Chuan did not stop, but instead wrote faster and faster.
The chalk scribbled across the small blackboard, making a rapid and crisp tapping sound.
[Question 3: The expression for h with respect to θ needs to be discussed in two parts. If the inclined plane is long enough, the effective kinetic energy of the ball along the inclined plane can be completely converted into gravitational potential energy. Substituting this into the equation, we get h(θ) = (8sin²θ)/(5).]
However, this expression only holds true if the length of the inclined plane is sufficient to bring the ball to a complete stop...
Halfway through writing, Han Chuan suddenly stopped and stared at the blackboard, rethinking his work.
Behind him, seeing Han Chuan stop solving the problem, Wei Jiqiu pushed up his glasses, his eyes filled with a hint of amusement.
As the person who set the question, he knew where the difficulty lay.
The solution to the third question is neither a straight line nor a simple parabola, but rather the graph of a sine square function truncated by geometric constraints.
This is also the most difficult part of the whole question. Han Chuan stopped writing when he got to this part, probably because he was stuck.
However, the fact that this student from Class 09 could complete the first and second sub-questions so quickly was already quite surprising to him.
For a regular class like this, who had never had any competition training experience before, it was quite impressive that they were able to solve the first two sub-questions of a semi-final competition problem in just a few minutes.
A very promising talent!
As for his earlier statement about solving the problem in 20 minutes, that was purely a matter of personal style and pressure.
After all, before coming here, Fang Jianyu had praised Han Chuan, this student, to the skies.
...
In the office, physics teachers Wei Jiqiu and Fang Jianyu sat quietly to one side, watching Han Chuan stand in front of the blackboard solving a problem.
Starting with the third question that got stuck, time slowly passed.
One minute.
Two minutes.
Five minutes.....
Just when Wei Jiqiu and Fang Jianyu thought Han Chuan had run out of ideas and couldn't solve the third sub-question...
Han Chuan, who had been standing in front of the blackboard pondering for a while, suddenly moved.
He quickly picked up the blackboard eraser from the chalk basket and wiped away all the steps of the solution to the third sub-question that he had written before.
Immediately, the chalk in his hand landed on the clean blackboard with a swish.
【E_eff = Ek·sin²θ= 8 sin²θ...】
【sinθ cosθ≤ 1.25→(1/2) sin2θ≤ 1.25→ sin2θ≤ 2.5....】
Two functions, one critical angle, three key points...
Then he drew a simple diagram next to it, labeling the h values when θ is at 0°, 20.7°, 45° and 90°, connecting them into a smooth curve.
This is a sin²θ rising curve starting from θ≈ 20.7°, with a jump at 20.7° (from 0 to 0.2m), and then smoothly rising to h=1.6m when θ=90°.
After writing the last word, Han Chuan put the chalk back in the trough and turned to look at Teacher Wei Jiqiu.
"Teacher Wei, I'm done."
The silence in the office lasted for a few seconds. Fang Jianyu, holding his thermos, stared blankly at the solution steps for the third question on the blackboard.
He had seen Han Chuan solve a problem in two different ways during the morning physics class, and he felt that this was the limit.
But now, Fang Jianyu realized that he had underestimated the student in front of him.
However, he didn't say anything, but turned to look at Wei Jiqiu beside him, his eyes filled with a hint of smugness and mockery.
He wanted to see Old Wei's reaction!
Meanwhile, the physics teacher who only taught the top classes had somehow gone from sitting to standing.
He stared at the solution steps for the third question on the blackboard for a long time until Fang Jianyu nudged him and asked.
"Old Wei, what do you think? The Han Chuan I mentioned isn't any worse than those top students you've been mentoring, is he?"
Coming to his senses, Wei Jiqiu nodded seriously: "It's really not bad!"
As he spoke, he stepped forward, picked up a piece of chalk from the chalk basket, and put a checkmark next to the answer.
"Perfect score!"
Immediately, he pushed up his glasses, looked at Han Chuan, and said, "In the constraint analysis of the third question, you didn't calculate the length of the inclined plane. You directly assumed that the inclined plane was long enough, and then used the constraint conditions to find out the effective interval."
"While this method may not be as rigorous and detailed as the standard answer, it is much simpler, more concise, and efficient!"
"Did you come up with this problem-solving method yourself, or did someone teach you?"
Han Chuan thought for a moment and said, "I guess you could say I came up with it myself, but I read it in a book."
"Which book?" Old Wei asked quickly.
"I've been reading 'Theoretical Mechanics' compiled by Mr. Zhou Peiyuan recently," Han Chuan said honestly.
He couldn't say that Mr. Zhou Peiyuan personally taught him, so this was the only way he could say it.
"Good, very good!"
Wei Jiqiu paused for a moment, then nodded and said, "Okay, you'll learn about competitions from me from now on."
As he spoke, he looked at Fang Jianyu and said, "Xiao Fang, give him the test paper. Let's assess his abilities and see where he has strengths and weaknesses!"
"I personally designed his training plan!"
.....
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