# Formulas converting Potential Energy to Kinetic Energy

• magnetics
In summary, the ball falls 1 meter in a vacuum with no resistance and its KE at impact is equivalent to its PE at its starting point.
magnetics
Gold Member

## Homework Statement

A 5kg ball (m), is to be dropped 1 meter (h) in a vacuum with no resistance in the Earth's gravity (g).

Use the speed and motion equations to show that its KE at impact is equivalent to its PE at its starting point.

PE = mgh
KE= 1/2mv^2
S = vt + 1/2gt^2
S = vt

## The Attempt at a Solution

Since h = 1
PE = mg

To calculate KE, we need to calculate the time taken to fall 1 meter and hence calculate the velocity.

h = 0 + 1/2gt^2
t^2 = 2/g

S = vt
v = h/t
therefore
v = √g/2
and
v^2 = g/2

KE = 1/2mv^2
KE = 1/2 x mg/2
KE = mg/4

I've now done this a few times and by the above calculations, the KE is 1/4 of the PE.

There has to be an error somewhere in my calculations, hopefully someone help enlighten me :)

magnetics said:
S = vt + 1/2gt^2

I've now done this a few times and by the above calculations, the KE is 1/4 of the PE.

There has to be an error somewhere in my calculations, hopefully someone help enlighten me :)

Try using (vt)2 = (v0)2 + 2aΔx. Your 1/4 problem should be gone then

magnetics said:

## Homework Statement

A 5kg ball (m), is to be dropped 1 meter (h) in a vacuum with no resistance in the Earth's gravity (g).

Use the speed and motion equations to show that its KE at impact is equivalent to its PE at its starting point.

## Homework Equations

PE = mgh
KE= 1/2mv^2
S = vt + 1/2gt^2
S = vt
I remember something like s = s0 + v0t+ 1/2 a t2
And in this case s0 is 1 m, v0 = 0 and a = -g

Since h = 1
PE = mg
...
h = 0 + 1/2gt^2
t^2 = 2/g
both OK (if interpreted somewhat liberally)

"S = vt" is NOT ok, though. This is only if v is constant, and v is not constant: there is a constant acceleration (namely -g)

"v = h/t" idem, and the "therefore v = √g/2 and v^2 = g/2" needs to be reviewed.

(What you want here is something like v = v0 + a t with v0 = 0 and a = -g )

" KE = 1/2mv^2 " true again, now we only need a better v .

2 people
BvU said:
I remember something like s = s0 + v0t+ 1/2 a t2
And in this case s0 is 1 m, v0 = 0 and a = -g

both OK (if interpreted somewhat liberally)

Thanks BvU.

v = at makes all the difference. You are correct, during the fall, a is constant, not v!

What do you mean by (if interpreted somewhat liberally)?

s = s0 + v0t+ 1/2 a t2 ; fill in
(in a coordinate system where up is positive, so g = -9.81 m/s2):
0 = h - 1/2 9.81 t2 bring to other side:
1/2 9.81 m/s2 t2 = h

The liberties are swapping s and s0 and the sign of g.
You have the benefit of the doubt (it comes out all right this time, but is that by coincidence?):
• I can see you take g = +9.81 m/s2, because you end up with t2=2h/g
-- also missed a factor h there: h is 1, so the value isn't the issue but you need the dimension of length to get s2 --
• but if you take g positive, you should really take h negative, )
but mistakes with signs and dimensions are made very easily, so be careful. Better to work explicitly and conscientiously.

Last edited:
BvU said:
[*]I can see you take g = +9.81 m/s2, because you end up with t2=2h/g
-- also missed a factor h there: h is 1, so the value isn't the issue but you need the dimension of length to get s2 --
[*]but if you take g positive, you should really take h negative, ) [/LIST]but mistakes with signs and dimensions are made very easily, so be careful. Better to work explicitly and conscientiously.

Just a note - if you're solving for t2, then making h negative if you have gravity as positive is totally fine. However, if you're solving for t, just a hint that you can't take the root of a negative expression and that you can do something about it.

## What is potential energy?

Potential energy is the energy that an object has due to its position or state. It is stored energy that has the potential to do work.

## What is kinetic energy?

Kinetic energy is the energy that an object possesses due to its motion. It is the energy that is required to accelerate an object to a certain speed.

## How can potential energy be converted to kinetic energy?

Potential energy can be converted to kinetic energy by allowing the object to move and harnessing the energy of its motion. This can be done through various methods such as gravity, springs, or chemical reactions.

## What is the formula for converting potential energy to kinetic energy?

The formula for converting potential energy to kinetic energy is KE = 1/2 * m * v^2, where KE is kinetic energy, m is the mass of the object, and v is the velocity of the object.

## Can potential energy and kinetic energy be equal?

Yes, potential energy and kinetic energy can be equal. This can happen when the object reaches its maximum height and has no more potential energy, but has gained an equal amount of kinetic energy due to its motion.

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