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I was thinking about this and couldn't really figure it out.

You are standing on a cliff and you have two identical balls. In this case, air resistance is to be ignored. The only thing different is that you throw the ball horizontally at different speeds. Lets say you threw ball one at a speed of 5m/s and the other ball at 10m/s

Will the balls land at the same time? Will the balls have the same speed when they land?

My thinking:

Since air resistance is ignored, I know that the horizontal component will be constant throughout. However, I wasn't really sure how no air resistance will affect the behavior of the ball. Could someone please answer the two question above?

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  • $\begingroup$ There are a couple of related question on this site that have some interesting answers. You are, of course, correct that the horizontal component of velocity for each ball is (ideally) constant since the gravitational force vector (due to the Earth) has no horizontal component. However, there is a very feeble gravitational force along a line joining the two balls due their mass. So, while the effect is too tiny to worry about for, say, two identical cannon balls, it is nonetheless there even if insignificant. $\endgroup$
    – Alfred Centauri
    Nov 2, 2018 at 3:28
  • $\begingroup$ Possible duplicate of 2 balls falling hit the ground at the same time $\endgroup$
    – sammy gerbil
    Nov 7, 2018 at 15:42

2 Answers 2

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In this case the balls will land at the same time indeed. First: The horizontal component have no effect over the vertical one at all. Since the velocity in $\vec{y}$ is the only responsable for the fall time and they are initially the same, both balls are going to have the same vertical speed at any instant and, thus, land together.

The overall speed wont be the same though, because it's the vector sum of the $\vec{x}$ and $\vec{y}$ components.

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Since the balls start at the same height and at the same vertical speed (0 m/s), they will hit the ground at the same time. The vertical velocity and position only depends on the gravitational attraction to the Earth and does not depend on the horizontal speed.

However, since one ball is faster than the other, the faster ball will travel a greater horizontal distance when it hits the ground.

You can think of both balls trajectories as upside-down parabolas, with the faster ball trajectory being more stretched in the horizontal direction.

As for the final speed of the balls, they will have the same vertical velocity when they hit the ground, but given that one ball as a greater horizontal velocity than the other, the ball with the greatest speed will be the one with the greatest horizontal velocity.

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