Which ball has the greatest speed just before hitting the ground?

Question

3 rocks of equal mass are thrown with identical speeds are thrown from top of building. X is thrown vertically down, Y is thrown vertically up, Z is thrown horizontally. Assume air resistance is negligible.

Answer 1

Ignoring air resistance, we are only interested in the initial and final energy states.

Initially KE = mv^2 / 2 for every ball.
Initially PE = mgh

Finally PE = 0
Finally KE = mv^2 / 2 + mgh for every ball

The magnitude of the velocity is the same in each case.

Answer 2

It is either ball X or ball Y. Ball Z is definitely out. However, which of the two balls? X or Y? Well, you will have to provide more information in order for us to determine which of the two balls will have the greatest speed as it hits the ground. Somebody could have thrown ball Y so high that when it comes back and hits the ground it will have much greater speed than X which was thrown verticallty downward. Or ball X was thrown with such a great speed downward while ball Y was thrown upward with only a very small upward velocity, which will definitely make ball X final speed greater than that of ball Y. So it depends on the initial upward velocity of ball Y and the initial downward velocity of ball X. teddy boy

Answer 3

If the initial speed is U, {directions differ for all the three},
Assuming no air resistance, and considering the vertical motions of the three,

For both X and Y,

V^2 = U^2 + 2g H
Though Y has a velocity ─ U, negative sign showing that it moves upward, since the formula contains U^2, both for Xand Y, V^2 and hence V is the same.

For Z,

the vertical speed is v^2 =2g H but its horizontal speed is U.

The resultant speed is V^2 = U^2 + 2g H

Thus all the three has the speed = √ [U^2 + 2g H]

Since U and H are the same for all the three, their SPEEDS are the same.

Answer 4

Everybody's making this way too complicated. ALL THREE rocks will hit the ground with the SAME SPEED, due to conservation of energy. You don't even need to analyze up, down or sideways.

At top of building: rock's total energy = KE + PE = mv_top²/2 + mgh

At bottom of building: rock's total energy = KE + PE = mv_bottom²/2 + 0

By conservation of energy:
mv_top²/2 + mgh = mv_bottom²/2

v_bottom = sqrt(v_top² + 2gh)

v_bottom is the same for all three rocks!

It is true that the _vertical_ velocities of X and Y will be greater than the _vertical_ velocity of Z. But that is not what the questions asks. The question just asks about the speed (regardless of the angle at which it hits).

Added: Sandy G says my answer is wrong "because we don't know how much energy is applied to the rock when it is 'thrown'." But to quote from the question, all 3 rocks are "thrown with identical speeds," and they have equal masses. To me, that means the amount of energy applied to each is mv²/2. We don't know the numerical value of mv²/2, but that doesn't matter. The fact that all 3 rocks start with the equal energy, means they all end with equal energy.

Answer 5

This question cannot be answered without additional information

I need to know the initial velocity of the rocks. I also need to know the height of the building.

I can pretty much rule out the rock thrown horzontally, though. It comes down to the two rocks thrown in the vertical plane. But, like I said there are too many variables to figure this accurately.

Sorry about that,

Stevo.

OK, I don't get it. You asked to figure out this question. It cannot be figured out, just assumed. So, why is it that I have all these thumbs down ratings? Can you that voted thumbs down actually PROVE that I am wrong? I'd be willing to bet you can't.

I can go through all of the reasons why you can't figure this if you want. All this talk of the rock thrown up will be going the same speed down as it passes the point where thrown is not accurate. Sure it will be CLOSE, but we don't want close. Never mind the fact that if it were going the same speed down, as up, it isn't thrown from that point, it is still free-falling.

Sure can throw in a few formulas to make my argument more compelling like V1 + 1/2 GT2 where you can substitute D for T If you knew the distance, but you don't. If the height of the building is tall enough, all THREE rocks would reach TV and be travelling at the same rate of speed. If the building is only a 10 story building terminal velocity (TV) would not be a factor.

This is still a very interesting question, and I would like to find out the answer as much as you, but we can't just go assuming things in a mathmatical equation, and that is what all of you are doing if you say one answer over another.

Answer 6

You cannot tell, because of the word "thrown"

Gravity accelerates at a constant rate of 32 feet per second per second, so the speed at which the object hits the ground depends on the initial speed and the length of the fall.

Firstly, mass doesn't matter. Small rocks and big rocks dropped from the same height reach the same speed when they hit the ground. (Galileo proved this)

A rock DROPPED straight down and a rock thrown horizontally will hit the ground at the same vertical speed because they are falling the same distance. The rock thrown up in the air effectively drops further and therefore will be moving faster when it hits the ground.

BUT, if the rock X is THROWN vertically down, it already has some velocity before it starts its drop and will be moving faster when it hits the ground. As we don't know how hard it is thrown down, we cannot tell whether it will be going faster or slower than Y.

Added: The answer from RickB is wrong, because we don't know how much energy is applied to the rock when it is "thrown". The rock thrown upwards falls from a far greater height because of the energy added to it by the upward throw.

Answer 7

The ones that are thrown vertically will have the highest speeds. The one thrown horizontally will have a lesser speed.

X has a speed which will increase with gravity starting at the height of the building...
Y will proceed up, then accellerate to the same speed it started at by the time it reaches its starting point, (so at the top the of building again) and then will be accelerated the same as X.
Z will have no vertical velocity to begin with so will only accelerate with gravity and will not make it to the same speed as Y or X. Using vectors you can add in the initial velocity horizontally and the vertical velocity, but it will never add up to the full velocity of X or Y.

Answer 8

Edit: READ!

I am deleting this post on the basis that the others are right. I knew based on the conservation of energy that the kinematic equations weren't working, and I finally figured out why. Those of us who assumed we could use these were forgetting one important fact. Even though X and Y have initial vertical velocities, they are in the air *less* than the Z, because of this velocity. This means Z has more time to speed up before it hits the ground.

So those of them who solved it based on conservation of energy are correct. They all hit at the same time.

Answer 9

The faster-thrown of X or Y.

Answer 10

X and Y hit the ground with the same speed, but X reaches the ground before Y.

Z reaches the ground some time between X and Y, but hits the ground with a lower speed than X or Y.