1) The initial height of the roller coaster must be 5.1 m
2) The height of the roller coeaster must be 5.5 m
Explanation:
1)
We can solve this problem by using the law of conservation of energy.
In fact, in absence of frictional force, the total mechanical energy must be conserved, so we can write:
[tex]KE+PE = KE'+PE'[/tex]
where
KE = 0 is the initial kinetic energy, at the top (the car starts from rest, so it has zero speed and so zero kinetic energy)
PE = mgh is the initial potential energy, where
m = 8000 kg is the mass of the car
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
h is the initial height
[tex]KE'=\frac{1}{2}mv'^2[/tex] is the kinetic energy of the car at the bottom, where
v' = 10 m/s is the speed of the car at the bottom
PE' = 0 is the potential energy at the bottom (where h = 0)
Re-writing the equation with all the values and solving for h, we find the initial height of the track:
[tex]h=\frac{v^2}{2g}=\frac{10^2}{2(9.8)}=5.1 m[/tex]
2)
In this case, we also have to take into account the energy lost due to friction. So the equation of conservation of energy becomes
[tex]KE+PE = KE'+PE'+W[/tex]
where
W = 32,000 J is the work done by friction
Re-arranging the equation and solving again for h, we find:
[tex]mgh = \frac{1}{2}mv'^2 + W\\h=\frac{v^2}{2g}+\frac{W}{mg}=5.1+\frac{32,000}{(8000)(9.8)}=5.5 m[/tex]
Learn more about kinetic energy and potential energy:
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