# basic physics - speed of a falling stone

Hi

Take this question: Gallileo drops a stone from the leaning tower of Pisa which is 45m high. At what speed does the stone hit the ground. (assume no air resitance)

In the book i am reading 2 different approaches to answering this question have been covered

1) basic acceleration formulae
e.g. v^2 = u^2 + 2as
I'm fine with this approach

2) relationship between gravitational energy at top and kinetic energy at bottom
this approach is saying that being as energy is conserved, the gravitational energy at the top will be the same as the kinetic energy at the bottom.

I'm less confident with the second approach. I presume this approach is saying that by the time the stone hits the floor, all of the grativational energy will have been transferred to kinetic energy because, on the floor, the potential energy from gravity = 0. I presume as the stone falls it will have a combination of both GPE and KE but when its back on the floor again all of the GPE that there was the top will now be KE. So you can then use the formula for GPE and KE, cancel out the mass and get the speed when stone hits the ground

If my assumption is right, you can only use the second approach to get the speed when the stone hits the ground because at any other point along the journey from top of tower to floor there is some combination of KE and GPE which i don't know how to work out yet

Many thanks
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Commented:
HI,
This may surprise you both solutions assume that all energy is conserved. So all the gravitational PE is converted to kinetic energy and none is lost due to friction.

An objects gravitational potential energy is due to its position in a gravitational field. As it falls, this potential energy decreases and the kinetic energy increases. So, yes, during this point it will have both PE and KE. When it hits the bottom, all the PE is lost and has been converted to KE.

To take this into consideration, as mentioned before, the GPE deals with an object's position and in most cases we take the surface of the earth to mean no GPE. It doesn't have to be.

So the gravitational PE is the difference between the top and the bottom that you are looking at; we don't need to consider the bottom to be zero GPE. If the difference between the top and bottom is a height, H, and the vertical position of the object between that height is y, then the total energy, mgH, is the sum of the potential energy and the kinetic energy at y. This can be expressed by the formula seen in the image.

Hope that helps.
David
formula.png
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Author Commented:
many thanks
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Commented:
When I was learning all this stuff and I couldn't "Get It" as in understand or visualize what is actually happening, I would do the math...  solve the formulas and then just keep doing it over and over and then it sinks in.

For this question it might be good to pick arbitrary points along the path of the stone ( say 20m above ground) and then solve for GPE and KE using formulas and then choose another point and do the same. Just keep doing it until it becomes clear to you.

At some point you need to just trust the formulas and then let your brain ease into comprehending. Just do the work. Understanding will follow.
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