Skiers on Different Sloped Hills?

[imaweinerz]

Skiers on Different Sloped Hills?

Homework Statement

There are 3 skiers , each going down hills of the same height, same x and y starting and ending points. There is no friction on the hill, and each skier starts from rest.
Hill 1 is a concave-up curved slope, dropping quickly and then dropping less rapidlybefore the end point.
Hill 2 is a straight line from the start and end points (linear).
Hill 3 is a concave-down curved hill, dropping slowly at first and then dropping more rapidly before reaching the end point.

Which skiier has the shortest time to reach the bottom, skier 1,2,3 or do they all have the same time, and why?

Homework Equations

kinematics equations. Ones I could think of were v=at, v^2-v(0)^2 = 2ax, etc.

The Attempt at a Solution

I first thought that skier 1 (the concave up skier) would reach the bottom in the shortest time because he has the highest initial acceleration. I know that all skiers have the same final velocity due to no friction and conservation of energy. My current thinking is that all three skiers have the same average acceleration, then they must take the same time to reach the bottom. Thanks for the help!

 

[guguma]

They reach the end points at the same time. You can think of it like this:

The potential at the problem is the gravitational potential, which leads to a path independent vector field.

y=x path would be more comfortable though, one of the skiers on the other paths have a chance to break their neck before the finish.

 

[alphysicist]

Hi guguma,

They reach the end points at the same time.

I don't believe that is correct. One skier is going at a higher speed for most of the path, and another is going at a slower speed for most of the path. Their final speeds will be the same, but the time to reach that final speed will be different.

 

[imaweinerz]

my thinking now is using the equation v = v(0) + at

All skiers start from rest (v(0) = 0), and all end with the same terminal velocity, due to conservation of energy.

I drew the hills, and noticed that in the middle of each path, the tangent was equal to the linear path, which to me indicated that a(avg) is the same in all 3 cases.

If a is the same, then by the equation (since all other variables are equal in v = v(0) + at), then time must be equal as well.

Is this reasoning correct? Thanks, I really appreciate the help

 

[alphysicist]

Hi imaweinerz,

my thinking now is using the equation v = v(0) + at

All skiers start from rest (v(0) = 0), and all end with the same terminal velocity, due to conservation of energy.

I drew the hills, and noticed that in the middle of each path, the tangent was equal to the linear path, which to me indicated that a(avg) is the same in all 3 cases.

No, I don't think you can say that. Remember that the average acceleration is the change in velocity over change in time. The last case will have a smaller change in velocity over a certain time, and the first case will have a larger change in velocity over a different time. So determining the average acceleration involves the time of travel, and therefore I think you need to reason out the time aspect some other way.

If a is the same, then by the equation (since all other variables are equal in v = v(0) + at), then time must be equal as well.

Is this reasoning correct? Thanks, I really appreciate the help

In these kinds of problem, take the slopes to extremes, so the slopes make "almost" right angles. Let the concave up case start out almost straight down, and then go horizontal at the bottom. Let the concve down case start out almost horizontal, and then go to almost vertical at the end. Think about the distances they have to travel, and the speeds they are going at. Does that help?