NEW YORK—Coco Vandeweghe was serving for the match and a place in her first US Open semifinal, but she was also teetering on the brink. While she led top seed Karolina Pliskova by a set and 5-3, Vandeweghe’s backhand was doing its best to sabotage her chances. Over the previous 10 minutes, she had slapped nearly as many backhands into the net as she had hit over it. Now she was down 15-30 and on the verge of letting Pliskova back into the match.

The Coco of old might have panicked and lost her cool in this situation. She might have slammed her racquet or screamed in rage. And, truth be told, she had done both of those things earlier. But now wasn’t the time to let her anger out. Now was the time to bury it and get on with the job at hand.

What do you do when you can’t hit a backhand? You make the most of every forehand you get. Vandeweghe rifled two crucial forehands for winners to dig herself out of trouble and complete the 7-6 (4), 6-3 upset. When she raised her arms and looked to her player box, her coach, Pat Cash, couldn’t hide his amazement. He shook his head a little, as if he couldn’t quite believe how well his player had just handled the most pressure-packed moment of her career.

Afterward, Vandeweghe sounded the way Cash looked: pleasantly perplexed by how well their partnership is going.

“I think the biggest thing [he has done] is channelling my intensity and tenacity out on the court, and putting it into a singular focus,” she said.

But exactly how had he done this? How had he improved her mentality at the relatively late age of 25? Trying to explain that, Coco was at a loss.

“You’ll have to ask him how he’s been able to do that,” she said. “I don’t really know. Maybe it’s, like, some Jedi mind trick. I don’t know how he’s doing it.”

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Vandeweghe, Anderson are showing what a little mental coaching can do

Vandeweghe, Anderson are showing what a little mental coaching can do

Vandeweghe isn’t the only player who has been making sudden and unexpected mental strides at the Open.

Twelve hours earlier, on the same court, Kevin Anderson had been in much the same situation as Vandeweghe. For three hours and four sets, he had been the superior player in his quarterfinal with Sam Querrey. He had competed, as he had competed all tournament, like a man possessed. At the moment, when it comes to fist pumps and “Come on!”s, Rafael Nadal and Maria Sharapova have nothing on the big, and previously mild-mannered, South African.

But now Anderson was on the verge of letting Querrey back in. After saving a match point, Querrey had reached set point in the fourth-set tiebreaker. The Ashe Stadium crowd was doing what it does after midnight: roaring for the American, cheering for more tennis and baying for blood. But no one was going to stop Anderson. While Querrey retreated and played defense, Anderson went after his shots with the same controlled fury he had used all match, and he eventually won the point, the set and the match. When he looked to his player box, his wife, Kelsey, had a similar reaction to Cash’s on Wednesday: She smiled in amazement at her husband, who had just reached his first Grand Slam semifinal at age 31.

Querrey led their head to head coming in, but on this night Anderson raised his game at the right moments and never caved, no matter what the score. Trailing 2-5 in the first-set tiebreaker, he came back to steal it 7-5. Trailing 1-6 in the second-set tiebreaker, he came all the way back to level it at 6-6. And he hung on to win the decisive breaker 9-7. All in all, Anderson’s performance was the most fearsomely stubborn of the tournament so far.

It’s also the most surprising. Anderson has always been among the most even keel and poker faced of players. Win or lose, he takes off his baseball cap, rubs his hair, shakes hands with his opponent and walks off the court. Now, after 13 years on tour, he’s expressing all of the desire he had kept bottled up, and hitting the ball with a new single-mindedness and unflagging intensity.

“I feel like that’s something I have sort of added to my game,” Anderson said of his newly aggressive attitude. “You know those matches are tough. I feel like I just try to play each point as best as I can … I felt I did a fantastic job resetting.”

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On the one hand, Vandeweghe and Anderson are opposing personalities with opposing problems to overcome. She’s trying to rein herself in while he’s trying to pump himself up. But each has gone about it in the same way, with help from a sports psychologist—or, as Vandeweghe prefers, a “mental coach.”

“It’s not a psychologist,” she told the WTA website this summer. “I’m still not a fan of it. I told the mental coach this, also: ’Listen, I’m not a fan of what you guys do.’ He said, ‘That’s fine. You don’t have to be as long as it works.’ I was like, ‘Prove me wrong.’”

Anderson was less leery of the idea.

“I just felt it was a part of my game I could maybe just get a bit better at,” Anderson told Reuters. “So many people say it’s 90 percent mental. Everybody knows how to hit tennis balls, and we’ve been playing tennis so long that it’s tough to make huge changes.”

As Anderson says, for a sport that’s supposedly “all mental,” tennis players have been loath to admit that they might need help with that side of their games. Watching Anderson pummel his way to victory last night, John McEnroe also sounded a note of skepticism about how much fist-pumping and psyching yourself up can really do for you.

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Vandeweghe, Anderson are showing what a little mental coaching can do

Vandeweghe, Anderson are showing what a little mental coaching can do

For many of us who watch tennis regularly, there’s also a certain amount of disbelief about how much any player can change. We see too many pros have a good week—or a good month, or even a good year—and then revert back to old, bad habits.

What I’ve liked about Vandeweghe and Anderson at the Open is how much their new mental outlooks seem to be helping their decision-making. She has become clearer about what her strengths are and how to use them, while he shows no hesitation whatsoever about taking a rip at the ball.

Old habits could, and probably will, creep back in for both of them. But there are reasons to believe their improvements will stick in the long run. Vandeweghe is proving that her trip to the Australian Open semis this year was no fluke, and that she can play with anyone. Anderson is getting the most out of a game that has always been there.

Either way, their late-blooming runs through the draws have been highlights of this year’s Open, and proof that seemingly insurmountable weaknesses can be overcome. You just need a Jedi mind trick or two to do it.