- #1
Ebby
- 41
- 14
- Homework Statement
- What is the minimum initial speed of the rollercoaster such that it will complete the vertical loop without falling away?
- Relevant Equations
- F_net_centripetal = mv^2/R
G.P.E._initial + K.E._initial = G.P.E._final + K.E._final (assuming no friction losses)
The problem itself is easy. My question is regarding the proper use of inequality symbols.
I only need to do the first part to show where I am having the issue.
The forces I need to consider are the coaster car's weight and the reaction of the tracks acting on it. With the car at the top of the loop, the reaction must be greater than or equal to zero or the car will have fallen away from the tracks:
Ignoring the inequality for the moment (yes, this isn't very mathematical), I'll just take the case when:
The governing equation is:
When , I can solve for :
And now I consult my intuition. I visualise the coaster performing the loop and I "see" that I must reinsert the inequality like this:
But this isn't very satisfactory. Surely I can do this inequality stuff so that one step in the calculation properly follows another - without any tricks. Perhaps I should introduce the governing equation like so:
And then I could argue that because actually it must also be the case that:
But I'm not sure this is logically sound... Can someone explain how to do this so it makes sense to me?
I only need to do the first part to show where I am having the issue.
The forces I need to consider are the coaster car's weight
Ignoring the inequality for the moment (yes, this isn't very mathematical), I'll just take the case when:
The governing equation is:
When
And now I consult my intuition. I visualise the coaster performing the loop and I "see" that I must reinsert the inequality like this:
But this isn't very satisfactory. Surely I can do this inequality stuff so that one step in the calculation properly follows another - without any tricks. Perhaps I should introduce the governing equation like so:
And then I could argue that because actually
But I'm not sure this is logically sound... Can someone explain how to do this so it makes sense to me?