That ball is gone!
[For purposes of this calculation, assume that the escape velocity from Earth's surface is exactly 7 miles per second.]
"The pitcher looked at him and grinned, then heaved a mighty ball;
The echo of that fearful swat still lingers with us all.
High, fast and far the spheroid flew; it sailed and sailed away;
It ne'er was found, so it's supposed it still floats on today."
Assuming an exit angle of exactly 30 degrees [a line drive, not a pop fly], just how fast in miles per hour would the exit velocity off the bat have to be for the baseball to escape the Earth's gravitational pull?
The answer is 25200.
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1 solution
That's how I answered the question too. But upon reflection, is that reasoning really correct? Suppose we said that the exit angle was 0 degrees. Using the same reasoning, we would conclude that the ball could never escape (even at infinite speed) because its velocity has no "vertical component". But if the Earth is a sphere (let's say a perfect sphere), an exit angle of zero degrees would simply be tangential to the surface, and I don't see any reason why it couldn't escape if it had enough speed. See Newton's Cannonball
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I agree. Since gravity is a conservative force we have path independence. Thus all that matters is the kinetic energy given to the ball, which is not dependent on the launch angle, (other than ensuring that it is launched at a non-negative angle). So the answer should be 25,200 mph regardless of the angle given.
Even if you were to drill a hole through the center of the planet out through the opposite side, if you threw the ball down the hole at 25,200 mph it would eventually escape gravity on the other side.
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If you think about the prospect of the ball going to infinity, any local geometry near earth has got to be inconsequential (as long as it isn't running into solid matter).
If I made a mistake, Brilliant staff can change the answer, and I'll acknowledge my error. Mechanics isn't my strong point, i'm better with pure math.
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@Denton Young – O.k., I'll put a note in the report section to get the staff's attention. Intuitively you would think that the launch angle would make a difference but I think my conservative force argument is valid. I like the poem, by the way. It has the same meter as "Casey at the Bat" but obviously has a happier outcome than the original. :)
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@Brian Charlesworth – Mudville finally gets some joy
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@Steven Chase – Hahaha Yeah, they had been waiting even longer than Chicago Cubs fans. :)
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@Brian Charlesworth – It's actually from the sequel to Casey at the Bat. Here's the full poem.
The Mudville team was surely up against a rocky game; The chances were they'd win defeat and not undying fame; Three men were hurt and two were benched; the score stood six to four. They had to make three hard-earned runs in just two innings more.
"It can't be done," the captain said, a pallor on his face; "I've got two pitchers in the field, a mutt on second base; And should another man get spiked or crippled in some way, The team would sure be down and out, with eight men left to play.
"We're up against it anyhow as far as I can see; My boys ain't hitting like they should and that's what worries me; The luck is with the other side, no pennant will we win; It's mighty tough, but we must take our medicine and grin."
The eighth round opened- one, two, three- the enemy went down. The Mudville boys went out the same- the captain wore a frown. The first half of the ninth came round, two men had been put out, When Mudville's catcher broke a thumb and could not go the route.
A deathly silence settled o'er the crowd assembled there. Defeat would be allotted them; they felt it in the air; With only eight men in the field 'twould be a gruesome fray, Small wonder that the captain cursed the day he learned to play.
"Lend me a man to finish with!" he begged the other team; "Lend you a man?" the foe replied; "My boy, you're in a dream! We came to win the pennant, too - that's what we're doing here. There's only one thing you can do - call for a volunteer!"
The captain stood and pondered in a listless sort of way. He never was a quitter and he would not be today! "Is there within the grandstand here"- his voice rang loud and clear "A man who has the sporting blood to be a volunteer?"
Again that awful silence settled o'er the multitude. Was there a man among them with such recklessness imbued? The captain stood with cap in hand, while hopeless was his glance, And then a tall and stocky man cried out, "I'll take a chance!"
Into the field he bounded with a step both firm and light; "Give me the mask and mitt," he said; "let's finish up the fight. The game is now beyond recall; I'll last at least a round; Although I'm ancient, you will find me muscular and sound."
His hair was sprinkled here and there with little streaks of gray; Around his eyes and on his brow a bunch of wrinkles lay. The captain smiled despairingly and slowly turned away. "Why, he's all right!" one rooter yelled. Another, "Let him play!"
"All right, go on," the captain sighed. The stranger turned around, Took off his coat and collar, too, and threw them on the ground. The humor of the situation seemed to hit them all, And as he donned the mask and mitt, the umpire called, "Play ball!"
Three balls the pitcher at him heaved, three balls of lightning speed. The stranger caught them all with ease and did not seem to heed. Each ball had been pronounced a strike, the side had been put out, And as he walked in towards the bench, he heard the rooters shout.
One Mudville boy went out on strikes, and one was killed at first; The captain saw them fail to hit, and gnashed his teeth and cursed. The third man smashed a double and the fourth man swatted clear, Then, in a thunder of applause, up came the volunteer.
His feet were planted in the earth, he swung a warlike club; The captain saw his awkward pose and softly whispered, "Dub!" The pitcher looked at him and grinned, then heaved a mighty ball; The echo of that fearful swat still lingers with us all.
High, fast and far the spheroid flew; it sailed and sailed away; It ne'er was found, so it's supposed it still floats on today. Three runs came in, the pennant would be Mudville's for a year; The fans and players gathered round to cheer the volunteer.
"What is your name?" the captain asked. "Tell us your name," cried all, As down his cheeks great tears of joy were seen to run and fall. For one brief moment he was still, then murmured soft and low: "I'm the mighty Casey who struck out just twenty years ago."
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@Denton Young – Ah, I didn't know there was a sequel. Thanks for posting it. :)
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@Brian Charlesworth – You're very welcome. It's one of my favorites.
With an exit angle of 30 degrees, the vertical component of the velocity is exactly 1/2 the total velocity. This component needs to be 7 miles per second * 3,600 seconds per hour = 25,200 miles per hour. So the exit velocity has to be double that, or 50,400 mph.
EDIT: Upon further review, the answer should be 25200. Gravity, as Brian points out below, is a conservative force that provides path independence.