## Up And Down The Mountain

**One day at noon, Annie runs to the top of a mountain. She sits and ponders the meaning of life until the next day at noon at which time she runs down the mountain along the same trail that she ran up. Was she necessarily at some point on the mountain trail at the same time on both days? Prove your answer — Mr. Simmons**

## Your Thoughts…

I don't think she was at soem point on the mountain trail at the same time on both days. This is because it is highly unlikely that you'd run up a mountain the same speed you would run down. A day is roughly 24 hours, and if she waits exactly one whole day before she descends down the mountain, there isn't much of a possibility of being at the EXACT same spot at the EXACT same time on both days. What do you guys think?

-Valerie C.

Yeah Val overall i agree with you but you really have to think about this from many points of view. I mean, you have to know if she was running at the same speed, like you said, if she did stay on top of the mountain until the exact same time of day the next day (again, like you said), and think if you want to look at it mathematically or logically. Mathematically, if your start time is the exact same both times you start the trail and her speed is constant, then she will reach the middle point at the exact same time as the day before. But logically, when you're running up a mountain it will probably take you up to twice the amount of time it will take you to run down. This is obviously because running down a mountain is a lot less exhausting and much easier to do. The mountain's slope would actually help you continue faster than it did when running up. So logically, she will never be at any point at the exact same time as the day before. My gut says NO!

- Rafa R.

I agree with both Rafael and Val! xD. If you get all sciency about it then it is impossoble. The force of gravity is pulling a person down. So when she tries to run up, she is basically running againt the force of gravity. This causes her speed to decrease, because even though the amount of work being done is the same, the amount of force applied increases. It is much harder for her to go up than go down, so the amount of force she exerts increases. This causes her speed to decrease. While when she is running down the mountain, she is running with the force of gravity. Since the gravity is pulling her down and she is moving downward, she doesn't exert as much forcce going down as she does going up. Therefore, her speed increases. So I think it is no. She would not be at the same point at the same time. did that make sense? also, this is scientifically speeking. I'm not sure if gravity plays a role in this problem. Because like Mr.Simmons said about the graphing the jumping fish, all this other stuff (more compplicated stuff) doesn't come till physics. So I'm going to stick with no. :D

~shreeya

Wait shreeya but then with what you said about mr simmons then that means he probably wants the mathematical answer. That would mean that the equation would just be D= RT and you wouldnt have to care at all about the direction youre running. So if

were going to use mr Simmons rules then we have to say yes!

- Rafa R.

Yeah but what Mr.Simmons said doesn't come up till physics!!! ahhh! I'm not sure.

~shreeya