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;

Question

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?

Answer 1

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.

Answer 2

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