Kaptain_Mittens
  • Kaptain_Mittens
a roller coaster has a speed of 1.5 m/s at the highest point. It moves down to the lowest point 2 and rises to another peak at point 3. The points 1, 2, and 3 are at heights of 20m, 5m, and 12m respectively. Determine the speeds of the roller coaster at points 2 and 3 if frictional losses are ignored.
Physics
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SOLVED
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jamiebookeater
  • jamiebookeater
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Kaptain_Mittens
  • Kaptain_Mittens
Just for reference, the answers are: v2=17.2 m/s and v3=12.6 m/s, but how do you get these solutions?
Kaptain_Mittens
  • Kaptain_Mittens
@Michele_Laino
Michele_Laino
  • Michele_Laino
the situation of your exercise, can be represented by this drawing |dw:1450114906621:dw|

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Michele_Laino
  • Michele_Laino
now, let's consider the peaks 1, and 2, respectively. Since the total energy (kinetic energy plus potential energy) has to remain constant, then we can write the subsequent equation: \[\huge \frac{1}{2}mv_1^2 + mg{h_1} = \frac{1}{2}mv_2^2 + m{h_2}\] where the subscript \(1\) refers to peak #1 and the subscript \(2\) refers to peak #2
Michele_Laino
  • Michele_Laino
oops.. I have made a typo \[\Large \frac{1}{2}mv_1^2 + mg{h_1} = \frac{1}{2}mv_2^2 + mg{h_2}\]
Kaptain_Mittens
  • Kaptain_Mittens
ok so we have: 1/2(m*1.5^2) + m(9.81)*15m = 1/2mv^2 + m(9.81)*7
Kaptain_Mittens
  • Kaptain_Mittens
how do we calculate this without knowing the mass?
Michele_Laino
  • Michele_Laino
please we have: \(h_1=20\) and \(h_2=5\) so we can write this: \[\Large \frac{1}{2}m{\left( {1.5} \right)^2} + m\left( {9.81 \cdot 20} \right) = \frac{1}{2}mv_2^2 + m\left( {9.81 \cdot 5} \right)\]
Kaptain_Mittens
  • Kaptain_Mittens
oh yeah, my mistake
Michele_Laino
  • Michele_Laino
now, if I divide both sides by the mass \(m\), I can write: \[\Large \frac{1}{2}{\left( {1.5} \right)^2} + \left( {9.81 \cdot 20} \right) = \frac{1}{2}v_2^2 + \left( {9.81 \cdot 5} \right)\]
Kaptain_Mittens
  • Kaptain_Mittens
I got 22.2 m/s, which evidently is wrong. I'll double check my math.
Michele_Laino
  • Michele_Laino
I got \(v_2=17.22\)
Kaptain_Mittens
  • Kaptain_Mittens
yeah, I re-did my math and got 17.22
Kaptain_Mittens
  • Kaptain_Mittens
I can finish up the problem now, thank you very much!
Michele_Laino
  • Michele_Laino
:)

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