Help With Conservation Of Energy Concept

[bobbles22]

Hi guys,

Can someone help explain something to me.

In a situation of an asteroid heading straight towards the Earth, I'm trying to understand where the energy comes and goes (using just gravitational potential energy and kinetic energy). Consider the energy of the asteriod some distance away from the surface of the Earth, then just before impact.

I'm using the following equations:

Kinetic Energy = E_k = 0.5mv²

Gravitational Potential Energy = E_g = (GMm)/r

My question is that, ignoring friction/sound etc, initially the asteroid has a certain speed and thus a certain kinetic energy, simple to work out. It also has a gravitational potential energy based on G and the masses involved and the separation. However, as it gets closer, the force from the Earth increased, as the separation 'r' off the objects decreases, the gravitational potential energy increases. At the same time, as the force on the asteroid increases, we expect the speed to increase so in effect, both E_k and E_g are both increasing. If both are increasing, then where is the energy coming from.

Surely it should be E_early = E_k + E_g = E_later = E_k + E_g

Can anyone give me a basic idea of where I'm going wrong with this idea please.

Many thanks

Bob

 

[Doc Al]

Where you are going wrong is with your equation for Gravitational Potential Energy:

Gravitational Potential Energy = E_g = (GMm)/r

You left out the all-important minus sign in front:
GPE = -(GMm)/r

However, as it gets closer, the force from the Earth increased, as the separation 'r' off the objects decreases, the gravitational potential energy increases.

Using the correct expression, you'll see that the GPE decreases (becomes more negative) as objects approach and r decreases.

See: Gravitational Potential Energy

 

[jtbell]

Gravitational Potential Energy = E_g = (GMm)/r

You omitted a minus sign. In this situation, we consider E_g to be negative.

However, as it gets closer, the force from the Earth increased, as the separation 'r' off the objects decreases, the gravitational potential energy increases.

No, the gravitational potential energy decreases as the object approaches the Earth, the same as an object close to the Earth for which we usually use E_g = +mgh. Including a minus sign in your equation for E_g takes care of this.

 

[davenn]

just a though to put out there ( I am not a maths guy )

you haven't given an example of the size/mass of an asteroid for your calcs

I would consider that the difference in size of say an asteroid up to ~ 1km across compared to the size of the earth, the gravitational PE may not be a large factor compared to the KE of the incoming asteroid.
average incoming velocity of a meteor/asteroid ... 30 - 40 km / sec
acceleration due to gravity 9.81m/s² ... big difference, orders of magnitude

some one more knowledgeable than me will hopefully clarify more

Dave

 

[davenn]

Thanks Doc and JT

that gave me some insight too :)

Dave

 

[Nugatory]

I would consider that the difference in size of say an asteroid up to ~ 1km across compared to the size of the earth, the gravitational PE may not be a large factor compared to the KE of the incoming asteroid.
average incoming velocity of a meteor/asteroid ... 30 - 40 km / sec
acceleration due to gravity 9.81m/s² ... big difference, orders of magnitude

But look at the units... The acceleration is measured in meters per second per second, meaning that every second the speed increases by 9.8 meters/sec... Let that go for a few hundred seconds and you've built up some serious speed. (note: gravitational acceleration is only 9.8 m/sec at the surface of the earth; it declines steadily the further away from Earth you are).

It turns out that an object falling from infinity to the surface of the Earth will reach a speed of about 11 km/sec; that's its gravitational potential energy being converted into kinetic energy. It's a lot.