Equal Speeds at y = 0 for Frictionless Slides?

[abpandanguyen]

Homework Statement

Consider 4 frictionless slides described by the equations

(1) y = sqrt(x), (2) y = x, (3) y = 2x, (4) y = x₂. If you start at rest at y = h and slide down to y = 0, which statement regarding your speed v at y = 0 is correct?

Homework Equations

(1/2)mv² = KE
mgh = PE

The Attempt at a Solution

The answer to this question is that all the speeds are equal at y = 0

I know the question asks for speed and that speed is distance over time, so doesn't that mean their speeds should differ taking 1, 2, 3, and 4 into account?

 

[kuruman]

Speed is "distance over time" only if there no acceleration, i.e. when the object covers equal distances in equal times. I don't know what y = x₂ means, but if it describes a slide, there must be acceleration in case (4) also. Try conservaion of mechanical energy and see what that tells you.

 

[abpandanguyen]

o whoops I meant to put that as x^2

I solved for v using conservation of mechanical energy but all my v values came out rather different still. I think I might be doing it wrong...

 

[kuruman]

Can you show what you did and exactly how your v values came out wrong?

 

[abpandanguyen]

here's one example

1.
mgsqrt(x) = (1/2)mv²

sqrt(2g(sqrt(x))) = v

2.
mgx = (1/2)mv²
sqrt(2gx) = v

 

[kuruman]

In all cases the initial kinetic energy is zero and the final potential energy is zero. So the initial potential energy is equal to the final kinetic energy. That's conservation of mechanical energy. Now answer this question, if in all cases the mass starts at height h, what is its initial potential energy?

 

[abpandanguyen]

I think I am getting confused by the
y=sqrt(x), y = x etc.

so having that aside, if they all start from the same height, that means they should all have the same initial potential energy and thus same final kinetic energy and given this, when you write the equations out, they all have the same velocity?

is x representing the horizontal distance here?

 

[kuruman]

I think I am getting confused by the
y=sqrt(x), y = x etc.

so having that aside, if they all start from the same height, that means they should all have the same initial potential energy and thus same final kinetic energy and given this, when you write the equations out, they all have the same velocity?

Yes, they have the same speed.

is x representing the horizontal distance here?

Yes, it does.

 

[abpandanguyen]

thank you!
want to help me on my other recent post? >_<