How Does the Particle's Velocity Change on the Roller Coaster Track?

[Celso]

Homework Statement

How can I find $h_{2}$ in terms of the other variables knowing that the horizontal distance between B and C is 10m? (the particle is initially at rest)

Relevant Equations

$E_{i} = E_{f}$

I first found $v_{B}$ by $E_{p,A,B} = mgh_{1} = E_{c, B} = \frac{1}{2}mv_{B}^2 \therefore v_{B} = \sqrt{2gh_{1}}$
After this I made several failed attempts basically trying to find its final velocity so I could use conservation of energy. Spliting the velocity into its components never worked because the force in these components varies with the angle as it falls.

 

[.Scott]

Are we told anything else?
For example, is that curve supposed to be a cubic curve?

In the segment from B to C, is the particle supported by the track - or is it following a parabolic path?

 

[Celso]

That's all, as far as I know. This is actually an

Are we told anything else?
For example, is that curve supposed to be a cubic curve?

In the segment from B to C, is the particle supported by the track - or is it following a parabolic path?

That's all, the problem's statement is simply "the following picture represents the configuration of a falling objetc". It's actually a problem I tried to solve for another person but I couldn't figure out after an hour

 

[PeroK]

That's all, as far as I know. This is actually an

That's all, the problem's statement is simply "the following picture represents the configuration of a falling objetc". It's actually a problem I tried to solve for another person but I couldn't figure out after an hour

What are you trying to calculate? Why can't $h_2$ be any height you like?

 

[.Scott]

That's all, the problem's statement is simply "the following picture represents the configuration of a falling objetc". It's actually a problem I tried to solve for another person but I couldn't figure out after an hour

There seem to only two constraints on the segment from B to C. The first is that the particle rests on the track, so the particle cannot fall any faster than if there was no track there. That would be a parabolic path and would give you a maximum value for h2.
The other constraint is that B appears to be an inflection point - with the path never again rising to meet the tangent line at point B. So at its highest, the B to C segment will follow just below that tangent line - giving the minimum value for h2.

 

[.Scott]

@Celso :
Could you show us the entire problem as presented in the book?
It appears to be written in Portugese.

 

[Celso]

@Celso :
Could you show us the entire problem as presented in the book?
It appears to be written in Portugese.

yes, it's in portuguese. I don't have the file now (it's on my PC), apart from what I've written in my previous answer, it only asks to find:
•The velocity at B
•The x and y components of the velocity at B
•The height h2 knowing that the horizontal distance between B and C is 10m

The first two are elementar, maybe that's an indicator that there might be missing information

 

[Celso]

What are you trying to calculate? Why can't $h_2$ be any height you like?

because if the problem is consistent (which I'm not sure), $h_{2}$ can be written as function of the other given variables

 

[PeroK]

because if the problem is consistent (which I'm not sure), $h_{2}$ can be written as function of the other given variables

Unless the path from B to C is falling under gravity (no track), then there is no unique solution for the shape of the track from B to C, as @.Scott has said.