# Throwing a ball sideways vs. dropping it

• Romain Astie
In summary, the conversation discusses a physics conundrum regarding the time it takes for a ball to hit the ground when thrown at high speeds on a round Earth. The participants consider the basic physics reasoning that it takes the same time for a thrown ball to hit the ground as a dropped ball, but also explore the possibility of the thrown ball entering a slowly decaying orbit. The conversation also touches on the concept of orbital velocity and the Earth's curvature. Ultimately, the participants conclude that while the ball will fall at the same rate, the distance it needs to cover on a curved Earth surface will make it take longer to hit the ground.
Romain Astie
So here's my physics conundrum:
I am on a ROUND Earth, in a vacuum. If I throw a ball really fast (say 99.9% of the orbital velocity for the ball's altitude), and drop another ball straight down, which takes longer to hit the ground?
My basic physics reasoning would say that it takes the same time, since from the thrown ball's reference frame, the distance to the ground is exactly the same whether it is thrown around the Earth or dropped in one place. The thrown ball simply spirals around the Earth to the ground, but effectively falls the same vertical distance as the dropped ball.
However, if you throw the ball fast enough, couldn't you get it into what is essentially a very slowly decaying orbit, thus making the thrown ball hit the ground after the dropped one?
How do I reconcile these two seemingly conflicting results, and what is the math behind it?
This is not homework or anything, just a thought I had today.

The math in question is for the "central force problem".
Clearly if the ball tangential velocity were high enough, it would never hit the ground ... so the "equal times to fall to the ground" rule is only a rule of thumb and does not apply in all situations.

Romain Astie said:
My basic physics reasoning would say that it takes the same time
Applies only in a uniform field, which is just an approximation of the actual radial field for small areas.

Romain Astie said:
So here's my physics conundrum:
I am on a ROUND Earth, in a vacuum. If I throw a ball really fast (say 99.9% of the orbital velocity for the ball's altitude), and drop another ball straight down, which takes longer to hit the ground?
My basic physics reasoning would say that it takes the same time, since from the thrown ball's reference frame, the distance to the ground is exactly the same whether it is thrown around the Earth or dropped in one place. The thrown ball simply spirals around the Earth to the ground, but effectively falls the same vertical distance as the dropped ball.
However, if you throw the ball fast enough, couldn't you get it into what is essentially a very slowly decaying orbit, thus making the thrown ball hit the ground after the dropped one?
How do I reconcile these two seemingly conflicting results, and what is the math behind it?
This is not homework or anything, just a thought I had today.
Welcome to PF Romain!

If you throw the ball into orbit, it will never hit the ground. So, if you throw it at 99.9% of orbital speed, it will take a long time to hit the ground. But it will still fall at the same rate as a ball that is dropped. It is just that the Earth surface keeps curving so it keeps having farther to "fall". On the Earth surface at the equator, orbital speed is about 5 miles/second. A ball will drop 16 feet in one second whether it is dropped or thrown horizontally (assuming its passage through the air does not create lift). But every five miles the Earth curves 16 feet. So while the 5 mile /sec thrown ball falls 16 feet in that first second, it then has another 16 feet to fall, etc.

AM

Last edited:
Romain Astie and Simon Bridge

## 1. What is the difference between throwing a ball sideways and dropping it?

The main difference is the initial velocity and direction of the ball. When throwing a ball sideways, the ball has a horizontal velocity component in addition to the downward force of gravity. When dropping a ball, there is only a downward force of gravity acting on the ball.

## 2. Which method will make the ball travel further?

Throwing a ball sideways will make it travel further, as it has a horizontal velocity that will help the ball cover more ground before hitting the ground. When dropping a ball, the ball will only travel vertically downwards.

## 3. Why does a ball thrown sideways curve?

This is due to the Magnus effect, which occurs when a spinning object experiences a force perpendicular to its direction of motion. In the case of a sideways thrown ball, the spin causes a difference in air pressure on either side of the ball, resulting in a curving trajectory.

## 4. Is there a difference in the force of impact between throwing a ball sideways and dropping it?

Yes, there is a difference in the force of impact. When throwing a ball sideways, the added horizontal velocity increases the overall speed of the ball, resulting in a greater force of impact. When dropping a ball, the force of impact is solely determined by the height from which it is dropped.

## 5. Which method requires more energy?

Throwing a ball sideways requires more energy, as it involves not only lifting the ball to a certain height but also imparting a horizontal velocity. Dropping a ball only requires the energy to lift it to a certain height.

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