Help with an energy problem found on the Phet Website

[Rainy_Pass]

Homework Statement

A 1000 kg roller coaster begins on a 10 m tall hill with an initial velocity of 6m/s and travels down before traveling up a second hill. As the coaster moves from its initial height to its lowest position, 1700J of energy is transformed to thermal energy by friction. In order for the roller coaster to safely travel over the second hill, it must be moving at a velocity of 4.6m/s or less at the top of the second hill. What is the maximum height the second hill can be?

Homework Equations

KE = .5mv²
GPE = mgh
total energy = PE + KE + Th_e

The Attempt at a Solution

PE_max = Total energy
PE = 1000kg (9.8 m/s²) 10m = 98,000J
KE = .5 (1000 kg) 4.6² = 10,580 J

Energy left at the bottom of the roller coaster = 98,000J - 1,700 J = 96,300 J

96, 300 J - 10,580 J = 85,720J

85,720 J = 1000kg (9.8 m/s²) h

h = 8.75m

 

[haruspex]

The question is a bit strange, and I'm not sure whether it is a mistake or it is deliberately laying a trap.
It provides a maximum speed for going over the second hill, but asks for a maximum height. Do you not see something odd about that?

 

[jackie311]

The explanation looks reasonable, but it's not the correct answer according to the answer key. The answer key says that it's 7.3 m, but gives no explanation of how to get the answer.

 

[jackie311]

Need to use:

85720-1000(4.6)(4.6)/2=1000(9.8)h

The first answer already subtracted the KE at the top of the second hill to get to 85720J, so why would you subtract it a second time? I'm trying to figure out the discrepancy between what the answer key is giving for a correct answer (7.3m) and what Rainy_Pass got for an answer (8.5m). The only thing that I can figure out is that the amount of thermal energy needs to be accounted for as a percentage, not a fixed amount. My Math skills aren't good enough to figure out how to put that into the equations though.

 

[haruspex]

The first answer already subtracted the KE at the top of the second hill to get to 85720J, so why would you subtract it a second time? I'm trying to figure out the discrepancy between what the answer key is giving for a correct answer (7.3m) and what Rainy_Pass got for an answer (8.5m). The only thing that I can figure out is that the amount of thermal energy needs to be accounted for as a percentage, not a fixed amount. My Math skills aren't good enough to figure out how to put that into the equations though.

The OP forgot the initial 6m/s. This may have confused @mfig .
As I posted, the question is strange. The maximum velocity at the end imposes a minimum height on the second hill. To find the maximum height we don’t need that. I suspect the full question asked, or was intended to ask, the min and max heights.

Another puzzle is whether we are supposed to consider friction in the ascent. Why would it suddenly cease? Ignoring that, I get heights over 10m for both min and max, so it looks like friction should be considered to continue, but we don’t know how to allow for it. Proportional to height, perhaps.