Simona Halep, the second-ranked tennis player in the world, uses a simple, fluid, matter-of-fact motion when she serves. There’s nothing too unusual about it — nothing showy or special. Technically, it’s excellent. But for a long time, Halep had a problem: She was afraid of it. You can see, maybe, where the moment of doubt would creep in — not the conscious doubt, whatever voice she hears as she bounces the ball in preparation for the point. I mean the briefest hesitation, the moment she has a choice: How high should she — could she — reach? Right after she tosses the ball, after her right arm swings back and her left extends straight, after her back foot slides forward alongside her front and her knees bend, everything seems to pause. Her body forms a long slight curve, a barely balanced arc. Then, just before the ball hangs at its peak, she leaps up, whipping her racket around and striking the ball at full extension. The ball will never come off her strings with quite the force she might want. Despite her strong shoulders and muscled core, at 5-foot-6, she’s too short; the physics don’t work. But if she uses her legs to propel herself up, and if she throws the racket head higher, she can use the air to make herself taller. She has to hit the ball at just the point she begins to fall, at just the point that feels too high, too hard to trust. She has had to learn, as her coach said, “to believe in it.”

On the Sunday before the start of the Western & Southern Open in Mason, Ohio, during a practice session, two players were standing across the net as Halep served. One was Halep’s hitting partner for the day, Lucie Safarova, a Czech now ranked 15th in the world. The other, Eugenie Bouchard, stood behind Safarova against the court’s back fence. Bouchard, a Canadian ranked eighth, loosened her wrist by wagging her racket back and forth so quickly it became a black and yellow blur. It was 11 a.m., her practice time, and she was waiting, patiently but not quite calmly, for Safarova and Halep to leave. As soon as Halep walked over to the corner, Bouchard skipped up to the net and began to hit at the service line, quickly pulling the rally back to the baseline and increasing the pace and depth of her hard, flat shots. Within a short time, her pink “Just Do It” T-shirt was dark with sweat. On television, her groundstrokes can seem strange, even hitchy. She takes the ball so early, and from such a low position, that she sidearms her forehand, whipping her follow-through so that the racket head sometimes sweeps past her left hip. In person, though, what stands out is her awesome aggression. Her low position loads her leg with force, and her swinging arms twist her core. She leaps into the court, freely swinging, her loose ponytail flying behind her, and she throws herself into the ball. There is a fearlessness to this, an utter confidence. Bouchard expects to hit winners. Winners are what she has worked so hard for. She swings for them as if they are her due. “Pressure is a privilege,” she said recently, quoting Billie Jean King. But in the past few weeks, since Wimbledon, Bouchard has been learning that privilege may have costs.

Halep is 22. The U.S. Open gives her the chance to take the no. 1 ranking from Serena Williams. Her goal at the beginning of 2014 was to remain in the top 20. At the beginning of 2013, she was fighting just to win two matches back-to-back — and fighting herself just to reach the ball. In 2011, she made the third round of the Australian Open, but during her first-round match in 2012, she was so nervous that she found herself unable to move her feet at all. She won the 2008 French Open junior title, but since then, she has struggled with injuries and fear. What changed was psychological and physical — it was impossible to separate the two. She was losing all the time, so she had nothing to lose. She won six tournaments between last May and December — more than anyone on the WTA except Williams — and took out top players along the way. But they were all small tournaments. She had yet to show she could do it on a big stage. Even she doubted she could. In Melbourne, the 20th seed, Dominika Cibulkova, crushed her in the quarterfinals, 6-3, 6-0. Halep won only 10 points in the second set. “She was overwhelmed in that moment,” her coach, Wim Fissette, told me. “She was emotionally not ready. She was not sure that she deserved to be there [among the final eight players in the tournament], and she was not confident enough about herself to say, ‘This is where I need to be. I can be better, I can be better than top eight in a Grand Slam.’”

Bouchard is 20. Only two years ago, she was playing in junior tournaments. In the quarterfinals of the 2014 Australian Open, she faced former no. 1 Ana Ivanovic, who had just beaten Williams. Bouchard broke Ivanovic’s serve seven times and won. The Australian Open’s official website anointed her “Queen Genie.” A group of Australian fans (nearly all men) proclaimed themselves to be Genie’s Army. The men presented the gorgeous then-19-year-old with a stuffed animal after each match: a wombat, a kangaroo, a koala, and a kookaburra. Bouchard accepted the adulation as she did the success. She wasn’t intimidated by the pressure, she said. She wasn’t overwhelmed. She attacked the net with unusual relentlessness. She stepped inside the baseline and hit punishing returns. She hit every ball on the rise. When she made the semifinals, she was the only one who wasn’t impressed. “It’s something I’ve been doing since I was five years old and working my whole life for and sacrificing a lot of things for, so it’s not exactly a surprise,” she said after the match. “I always expect myself to do well.”

Fissette, who has coached four-time Slam winner Kim Clijsters, became Halep’s coach in February. Before they met for the first time in a hotel lobby in Paris, she wrote him an email apologizing for her poor English. He was struck by how shy she was. “She was really nervous,” he said. “It was all new. Someone from outside of Romania. It was out of her comfort zone.” He’d seen her play before, though, and had a sense of how great her game could be. He recognized her balance, her anticipation, her intelligence. He noticed how focused she was on changeovers, and how quick were her feet. But her game, he thought, was essentially defensive. If she was going to win majors, she was going to need a more offensive style. That meant any improvements to her game would have to begin with her mind. He wanted her to move up to the ball to cut the angles with her forehand instead of stepping backward. “If you’re confident, you can do that well. So it goes together.” He wanted her to use more of her legs in her serve. “I told her, technically, her serve was very good,” he said, “but she had to believe in it.” Shortly after they began to work together, Halep played a tournament in Doha, and during her first match struggled to beat a lower-ranked player, Kaia Kanepi. Outside of majors, on-court coaching is allowed in WTA tournaments, and Halep called Fissette onto the court every set. She listened to him, less for what he said than to see if she could trust him. She went on to win the tournament, beating three top-10 players along the way, including Agnieszka Radwanska, who had once defeated her so badly that Halep had thought, I cannot beat her ever. By the end of the Qatar Open, she was able to tell herself that she was better than the others, that she was the best. During practice, she’ll sometimes glance over at Fissette, and he’ll say a quick, quiet word, focusing on the good things in her game. He wants to give her confidence. “I don’t think that really happened in the past,” Fissette said. He tries to tell her how good she is, he told me, and then laughed. “I think she believed me.”

When Bouchard practices, her coach, Nick Saviano, paces in the backcourt, watching from different angles, sometimes pulling out his phone to film her. Frequently, he interrupts the hitting to correct her strokes. His intensity is obvious; it seems as strong as Bouchard’s own. They have been working together since she was 12. Saviano likes to point out that Bouchard has a complete game: She is strong on both wings, has a solid serve, attacks the net, and volleys well. But what impresses him most about her game — what drew his attention to her — wasn’t any special physical talent. It was her dedication and her drive. He likes to talk about her confidence, her thriving under pressure, the things that, “as a coach, you can’t teach.” On the court, that translates into an ability to not only swing freely in big moments, but also to play her own style regardless of the situation. Bouchard has played her best tennis on the biggest stages, the most prestigious and most exciting tournaments, where other players tend to tighten up, restricting their movements and altering their swing. She is aggressive and agile, and it can seem like she somehow dominates the space she’s in. She doesn’t change her strategy, regardless of the situation. Her game is, as Halep said when I asked her to describe it, “always the same”: Hit the ball early, and hit it hard. “She’s tall when she stands close to the baseline,” Halep said during Wimbledon. “It’s like you see just her on court.”

Halep’s best skill is her ability to adjust. “I don’t have a plan when I go on court,” she told me. She waits a few games to see what the other player is doing, then decides what strategy is most appropriate. Before she steps to the baseline to serve or receive — her return is one of her strengths — she knows exactly what shots she wants to play. “She is always, always thinking,” Fissette said. “She’s always thinking about how to win. And you see also, in the match, she’s changing her game.” Her game starts with her feet, which are quick and light on the court. They are never still, always dancing. She has perhaps the best footwork in the game, and an unusual ability to run down balls, partly because of her speed and partly because of her anticipation. She sometimes not only seems to know where her opponent’s shot is going before her opponent hits it — sometimes she actually does know. “I like to open the court, to open the angles,” she said. “I liked math at school, so I was good at finding the angles. Now I just want to play beautiful on court, you know? Sometimes too much, but it’s still good to have a little bit of imagination.”

Bouchard has been open about her single-minded dedication to tennis. The game is its own end. Still, she is beautiful, and marketable, and she knows it. She is participating in the creation of her public image. She tweeted a picture of herself wearing a robe in bed, her tousled hair framed by the pillow (caption: “Chillin like a villain #tired #selfie”), and came to a press conference dressed in a red kimono. The New York Times Magazine put her on the cover of its U.S. Open issue, in front of the words “BIG SHOT.” (Bouchard declined several requests to be interviewed for this story.) Bouchard’s contract with her agency, Lagardère, expires this year, and at Wimbledon, other agents followed her in a pack. She enjoys attention, and why shouldn’t she? She is 20 years old, and it is fun. Perhaps it also allows her to be sociable in a way the rest of her life does not.

“I don’t think the tennis tour is the place to have friends. For me it’s all competition,” Bouchard said at Roland Garros. Few players have very good friends on tour, but Bouchard takes it to a greater extreme. “She’s never speaking with anyone,” Halep said. “No hello, no anything.” Part of her image is her fierce independence, and what that means, in a day-to-day sense, is that time on tour can be lonely. She spends much of her time with her mother, Julie Leclair, who travels with her, and her coach. Leclair told The New York Times Magazine’s Susan Dominus that Saviano is “almost like a second father.” Bouchard teases him in press conferences and argues with him in practice, chaffing as a teenage daughter might. Bouchard’s real father is an investment banker; her parents are divorced. When she was 9, her father and two partners started a limited partnership to invest in her future earnings. This summer, he lost a four-year battle in Canadian courts to claim his investment in his daughter’s development as a tax write-off. Bouchard is close to her fraternal twin, Beatrice (all the Bouchard children were named after British royals), who insisted there was no jealousy between them. Still, familial affection can have sharp edges. Bouchard’s siblings call Genie, in an homage to Harry Potter, “The Chosen One.”

For much of her quick ascendance, Halep was barely known, even in tennis circles. Few reporters attend her postmatch press conferences. Even when she is among the highest seeds in major tournaments, she is often relegated to smaller courts. She says she doesn’t mind. “They have their own players from here, and they have also the stars,” she said. “Maybe I can say I am a little star, but still, I just came into the top.” For a time, she shied from attention. She is from a small town in Romania — Constanta, on the Black Sea. Her father runs a small milk and cheese factory, while her mother travels with her on tour. This spring, though, she made runs in several big tournaments, and at the French Open, she made the final, losing to Maria Sharapova in what was the best Grand Slam women’s final in the past decade. Back in Romania, Halep was startled to realize that in Romania she is famous. One day, when she was parking her car near her apartment in Bucharest, she was recognized by the construction workers repairing the street. People came up to her in the grocery store to say hello.

“It wasn’t very cool in the beginning,” Halep said. It was exhausting. “Everybody knows me. I have to know everybody.” Then, though, she was startled to realize she enjoyed it. People were kind. They congratulated her. They told her she made them proud to feel Romanian. At her home tournament, right after Wimbledon, she was always put on the main show court. Romanian legends Ilie Nastase and Nadia Comaneci were in the stands to see her win. The prime minister named her an ambassador of tourism, which meant she received a diplomatic passport she used to enter the United States in early August. “The security in the USA asked me, ‘Why you have diplomatic passport?’ I said, ‘I don’t know! It’s not my fault!’”

A month ago, when Bouchard stepped onto Centre Court at Wimbledon, most people — including, it’s probably safe to say, herself — expected her to walk off it as Wimbledon champion. In the semifinals, she had beaten Halep, taking advantage of her opponent’s limited mobility after Halep turned her ankle in the first set, and punishing her with hard, skidding groundstrokes. Instead of winning the final, though, Bouchard was crushed by Petra Kvitova, 6-3, 6-0, in one of the most dominant performances I’ve ever seen. No one was going to beat Kvitova that day. More startling is what happened a few weeks later, in Montreal — Bouchard’s hometown. The crowd was roaring with adulation. A contingent of Genie’s Army had flown up from Australia for just an occasion. Bouchard was facing a 113th-ranked American, Shelby Rogers. And almost before she could take her first breath, Bouchard found herself down 5-0. Bouchard summoned Saviano. “I want to leave the court,” she said to him, and covered her face with a towel. “I understand,” he said. “I understand.” Bouchard won the next set, 6-2, but then lost the third set, 6-0.

“I think where Simona is right now is where she belongs,” Fissette told me. “Top three. I believe she can be better. I believe, if things go better, she can be no. 1 one day.” When I asked him when, he said, “I think she can do it next year.” Halep has become more open, more talkative when she talks to the press. She speaks in a quick, clipped, flat monotone, and she is quick to laugh. Fissette marvels at the difference between her now and when he met her in February. “I feel she’s a different person now,” he said. But her own “dream,” Halep insists, is more modest than the ones he describes. What she wants most is to make the year-end WTA finals in Singapore. The top eight performers of 2014 qualify; she is, right now, in the second spot. I asked her why she was focused on that and not a Slam. “During the year, we have normal draw,” she said. “So if you lose, you go home. There, you lose, you can stay.”

Bouchard lost again in her first match in Cincinnati and struggled in the New Haven Open, the last tournament before the U.S. Open. It’s not clear how much to read into her losses in smaller tournaments. She long ago established a pattern of getting knocked out early, then performing well, as she said after her loss in Cincinnati, “when it counts.” What made these losses different was the attention given to them. That attention shades her a little differently than cameras do. There is something poignant about her hurry to win now — something that both is at odds with the image she projects and underscores it. “I just turned 20 three months ago and I feel old,” she said at the French Open. “So I want to be the best player I can be as quickly as possible, because one day I will wake up and I will be 30.” When Elle Québec asked Bouchard what she most feared, she answered that she was scared of aging.

In her first-round match at the U.S. Open — the first match of the tournament at Arthur Ashe Stadium, the massive main show court — Halep faced an American named Danielle Rose Collins. Collins, who qualified by virtue of winning the NCAA title, is the no. 2 player … at UVA. Halep quickly went up a break in the first set, but then you could see her tighten, visibly. Easy put-away shots clipped the net. Her forehand went wild, and the errors started to mount. She lost the first set in a tiebreak. She is still prone to getting frustrated, still nervous when she’s down. She will yell at herself or swipe her racket. “Always I have emotions. Everything I do, I have emotions. Before every match, I have emotions.” She is trying to learn to “put them away,” to focus on the single point at hand. There, I have to cross, she will tell herself. Then very fast on the right side, then maybe a drop shot. But that can be impossible. She doesn’t hide her feelings well. “Sometimes I am angry, yelling at myself,” she said. “Sometimes I give some courage to myself.” She is, despite her ascent, despite her experience and maturation, still sometimes overwhelmed. Even as she settled down against Collins, easily outplaying her in the second and third set, she had a tendency to race out to a lead and then make an easy play, letting her opponent back into the point, the game, the match before finally closing it out. It wasn’t a question of skill: Halep is one of the very best players in the world and Collins — though she can hit gorgeous winners — is not even ranked. The match that mattered was inside Halep’s mind. How high should she, could she, reach?

Bouchard comes in as the fifth seed, and she has a good draw. If the seeds hold, she could face Kvitova again in the quarterfinals — but there’s a good chance Kvitova won’t make it that far; the Czech has always struggled in New York. If Bouchard is playing as well as she’s played at Slams this year, she should have a relatively easy path to the semifinals, where she could face Williams. Bouchard won this morning, 6-2, 6-1.

The future is easy to predict only in retrospect. The boring truth is that what we know now is that Bouchard, like Halep, sometimes rises to the occasion and sometimes does not. We read meaning into moments. We think we can spin a narrative on the basis of whether, at one particular moment or another, a player hits a ball two inches in or two inches long. We think where it lands says something about who she is. The craziest thing is, for her to be a champion, she needs to think so, too.