Problem 3

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 think that it is possible that she can be at the same time time at the same point on the trail but it is not very likely due to the probable change in speed she has when she goes downhill. If she travels at a constant speed, then she will probably meet at least somewhere on the trail.


im first again!!
my guess: this is pretty confusing but i think she could have. because say that she runs half way up at 3:00 p.m. and then the next day, she runs half way down at 3:00. but many factors could affect her. say, she is tired or there is more wind than usual, or she trips and tumbles down the mountain

joseph l.

I think that because it asks if she necessarily at some point on the mountain trail at the same time on both days, we can just try to create a scenario in which this does not occur.

I think that yes, she was necessarily at some point on the mountain trail at the same time on both days because If she goes at the same speed, the spot will be halfway down the mountain.
Speed doesn't seem to matter, because if she goes faster (on the way down), the spot will just be higher, and if she goes slower, the spot shift lower. But it doesn't matter if she crawls her way up the mountain, or if she drives a car down, there'll always be a spot between the top and the bottom where she'll be at the same time both days.

Joseph, all your factors only affect her speed, but speed seems to be irrelevant in this problem. There's probably some other factor I overlooked, however. I'll be thinking about that.
-James D.

what do you mean speed doesn't matter? if she goes crazy slow going up and fast going down, then heck yea, it's gonnna affect the time.

joseph l

Thinking out loud here. If she took 8 hours to go up, then 4 to go down, would it be the same?

Well, since every hour she moves 1/8 of the way up, or 1/4 of the way down, then we can add/subtract that amount every hour.


1st hour:
Up:0 + 1/8= 1/8
Down:1 - 1/4= 3/4

2nd hour
Up: 1/8+1/8=1/4
Down: 3/4-1/4=1/2

3rd hour- This is where she would over reach. The person going down would travel 2/3 of the distance left between them (1/4) In the time it takes the other to go 1/3, so they end up meeting 4/12 of the way up, or 1/3 of the way. They each took the same amount of time to get there, so they are both the same amount of time after noon. In conclusion, yes, she would be at a same place at the same time regardless of speed.

Tucker W. (Hope I did this right…)

Another example… just to be sure.
Up- 20 miles per hour. 20/21sts of the trail
Down- 1 mile per hour. 1/21sts of the trail
The mountain trail is 21 miles long.
It's obvious that they'll meet after an hour.

Another way to think of this is: If she goes incredibly fast on the way up, and incredibly slow on the way down, the spot will be incredibly close to the top. If she does the opposite, the spot will be incredibly close to the bottom. And the closer the speeds are to one another, the closer the spot will be to the middle.

So if we can agree that speed in irrelevant, the answer should be yes, but other contributions are still welcome.
-James D.

that doesn't make any sense. the question says same time

joseph l

Speed really only affects at what point they overlap. Thinking of it as two people makes it easier to understand. Now, if they both start running up/down, they WILL meet at one point. And since they meet at that one point at the same time, the same time that they left the top, then yes, she would meet "herself". Understand?

-Tucker W.

She might not, because it's impossible to run up a mountain and run down a mountain at the same speed. Of course she's not gonna go at the same speed. Then she doesn't have to be at the same spot at the same time. So the answer is no, she does not!
-Yash T.
P.S. Tucker, it's not the same time after they leave their starting place, it's the same time of day that she has to meet on from the day before!

Tucker obviously knows this, and if you checked his explanation or mine, you would know.
Here, another explanation for those who need it.
Say she startes at 12:00 pm, and takes 6 hours to get to the top, and the distance to the top is 6 miles . she travels at 1 miles per hour. On the way down, she takes only 3 hours. She travels at 2 miles per hour.

1:00 pm both days.
Distance from ground:
Up- 1 mile (0+1)
Down- 4 miles (6-2)

2:00 pm both days.
Distamce from ground:
Up- 2 miles [(0+1)+1]
Down- 2 miles [(6-2)-2]
They're both in the exact same spot at exactly 2:00 pm both days, even though she went twice as fast on the way down.

Now we know that SPEED DOESN'T MATTER!
So the answer is yes, until someone can actually prove with an example that there can be an occasion in which there is no spot where she'll be at the same time both days. We've supported our claims with lots of examples. Maybe you ought to try doing that.
-James D.

thank you tucker. for helping me out. now i understand. at first i thought it was same place same time which might be possible.

joseph l

No problem Joseph. Yash you have to remember that she is constantly going over area that she went over the day before. It is only a matter of time before she meets a point where she was at that time the day before.

-Tucker W.

I guess thats true, at the begning she's at the same location. I can't really do much now, can I?You guys gave really good examples.
Teju K.

:( my karma level isn't getting any higher…

joseph l.

Sorry Teju, you really need to get in on it early ;-)

Why don't you draw a picture to help Yash? I'm too lazy to :O

@Joseph: Your what?

-Tucker W.

How do you check your karma level??
-James D.

Hey guys. I think you're right but since she doesn't start at the same place… wait. nope. you guys are right. It doesn't say she goes at different speeds so we should assume she will be at atleast one spot on the mountain at the same time both trips. Btw… Is this page hard to read for you guys? The font on my computer is super tiny! I can barely read this!
Katie B.

What internet browser are you using? Trying pressing Ctr and scrolling with the mouse (The thing in the middle)

-Tucker W.

click on the history button on the bottom and you can see your karma leevel.

joseph l

Tucker is right even if she is moving like a snail on the way down the faster time will catch up. there is no possible way (if she moves like she did in the problem) that she could not be at the same point at one time. The question is, whether or not she really went at a constant speed the whole time … and the meaning of life. :)

Mark F.

I guess maybe you guys are right… It does make sense now.
-Yash T.

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