Question

Two identical balls are thrown vertically upward. The second ball is thrown with an initial speed that is twice that of the first ball. How does the maximum height of the two balls compare?
The maximum heights of the two balls are equal.
The maximum height of the second ball is two times that of the first ball.
The maximum height of the second ball is four times that of the first ball.
The maximum height of the second ball is 1.41 times that of the first ball.
The maximum height of the second ball is eight times that of the first ball.

Answer 1

Seems to me like it would be twice as high, but no! Use conservation of energy:

1/2 mv^2 = mgh ---> h = v^2/(2g)

Now let v --> 2v, then h --> h' = 2v^2/g

So second ball travels 4 times higher.

Consider also, I assume you are on the Earth, the balls are in fact, once you release them, in orbit. The perihelion is the top of its orbit, maybe just a few feet off the ground depending on how hard you throw - But it's path (before it lands) is actually a very small arc of its entire orbit around the planet. with its aphelion a few feet above the earth on its far side- a very narrow but elliptical orbit. If the planet weren't physically in the way to block its path, it would revolve in an orbit.

Answer 2

Seems to me like it would be twice as high, but no! Use conservation of energy:

1/2 mv^2 = mgh ---> h = v^2/(2g)

Now let v --> 2v, then h --> h' = 2v^2/g

So second ball travels 4 times higher.

Consider also, I assume you are on the Earth, the balls are in fact, once you release them, in orbit. The perihelion is the top of its orbit, maybe just a few feet off the ground depending on how hard you throw - But it's path (before it lands) is actually a very small arc of its entire orbit around the planet. with its aphelion a few feet above the earth on its far side- a very narrow but elliptical orbit. If the planet weren't physically in the way to block its path, it would revolve in an orbit.

Answer 3

Seems to me like it would be twice as high, but no! Use conservation of energy:

1/2 mv^2 = mgh ---> h = v^2/(2g)

Now let v --> 2v, then h --> h' = 2v^2/g

So second ball travels 4 times higher.

Consider also, I assume you are on the Earth, the balls are in fact, once you release them, in orbit. The perihelion is the top of its orbit, maybe just a few feet off the ground depending on how hard you throw - But it's path (before it lands) is actually a very small arc of its entire orbit around the planet. with its aphelion a few feet above the earth on its far side- a very narrow but elliptical orbit. If the planet weren't physically in the way to block its path, it would revolve in an orbit.