If I went tubing with my little sister, and I have a larger mass, why would we both reach, in terms of energy, the bottom of the hill at the same time;

Originally I thought because we would both be accerlerating at the same rate but that doesn't explain the energy part does it?

But then I think it would be because the TE(top) of the hill = TE(bottom of hill) and when you set that equation up, mass drops out, correct?

Thank you for the detailed answer-I actually get it and understand it-thank you again!!

You are correct in thinking that the acceleration alone cannot explain why you and your little sister would reach the bottom of the hill at the same time, in terms of energy. The key concept here is the conservation of energy.

When you go tubing down a hill, there are two forms of energy that are relevant: potential energy and kinetic energy. Potential energy is the energy stored in an object due to its position, while kinetic energy is the energy of an object in motion.

As you go downhill, your potential energy is converted into kinetic energy. The higher you are on the hill, the more potential energy you have, and as you descend, this potential energy decreases while your kinetic energy increases. At the bottom of the hill, all your initial potential energy is converted into kinetic energy.

Now, if we assume that the total mechanical energy (potential energy + kinetic energy) is conserved along the way, then the total energy you and your sister have at the top of the hill will be the same as the total energy you both have at the bottom of the hill.

Since energy is conserved and assuming negligible energy losses due to friction and other factors, the total energy at the top of the hill is equal to the total energy at the bottom of the hill. This means that the sum of your potential energy and kinetic energy, compared to that of your sister, will be equal.

In the case where you have a larger mass, you will have more kinetic energy at the bottom of the hill, compensating for the lower potential energy compared to your sister. This extra kinetic energy makes up for the difference in mass and allows you both to reach the bottom of the hill at the same time in terms of energy.

So, in summary, when you set up the equation TE(top) of the hill = TE(bottom of hill), you're taking into account the conservation of energy and the fact that potential energy is converted to kinetic energy as you descend. Mass cancels out in this equation because it doesn't affect the conservation of energy principle in this scenario.

It is rather neat the answer to this,and you need to thank your teacher for asking this neat question.

Now you know what is moving you is Graviational attraction, which depends on your mass

Fg= GMearth*Massyou/radiusearth^2

so clearly, the Earth is pulling on You much harder than sis.

But what is really neat about this, is to compare what that differing force does on you and your sister.

Newtons second law:
Force=mass*acceleration or
acceleration= Force/mass

Your acceleration= Yourgraviationalforce/your mass= GMearth/radiusearth^2

and low and behold, your sister has the same acceleration down the hill. It isn't g, it would be g if you fell vertically, but is the same.

Now in terms of energy: Consider both of you having the same potential energy at the top of the hill, say PEorig.

That PEorinal becomes kinetic energy as you go down the hill.

Because the amount of work Gravity is doing (work= force*distance) is different on each of you, lets see what it does to each of your Kinetic energies.

workyou=force*distance=GMe*Myou*distance/radiusearth^2
and
worksister=GMe*Msister*distance/radiusearth^2

but kinetic energy gained has to equal this work done.

KEyou= 1/2 massyou*v^2=GMeMyou*distance/radiuearth^2
or v^2=2(GMe*distance/rearth^2)

but if you do the same thing finding the KE of your sister, guess what (do the math), for your sister
V^2=2*GMe*distance/rearth^2)

now since you went the same distance, you have the same velocity.