Physics, Energy Problems (confused!)

Question

1) a 2000 kg rollercoaster starts from a height of 55 m and an initrial velocity of 8 m/s. When it reaches the bottom of the hill (height of zero), what is its velocity?

2) For the rollercoaster problem above (m=2000kg), if the final velocity at the bottom of the hill was only 27 m/s instead of 34 m/s, how much energy would have been "lost"? (Lost mechanical energy is the Eother)

I got 427,000J for the 2nd probem, but according to my teacher it's 435,000J. Was that simply a mistake?

Answer 1

(1)

E₁ = mgh₁ + mv₁²/2 = 0 + mv₂²/2

v₂ = √[2(gh₁ + v₁²/2)] = ?

Plug in the numbers and get approximately 33.79 m/s, which can re rounded off to 34 m/s.

(2)

ΔE = E₂ - E₁ = (0 + mv₂²/2) - (mgh₁ + mv₁²/2) = m[(v₂² - v₁²)/2 - gh₁]

In this part of the problem,

v₂ = 27 m/s

ΔE = m[(v₂² - v₁²)/2 - gh₁] = 2000[(27² - 8²)/2 - 9.8(55)] J = -413,000 J

I get a TOTAL energy loss of 413,000 J. This makes sense, because at the top, the total energy is

E₁ = m(gh₁ + v₁²/2) = 2000[9.8(55) + 8²/2] J = 1,142,000 J

At the bottom it is

E₂ = m[0 + v₂²/2) = 2000(0 + 27²/2) J = 729,000 J

The loss must be the difference between these two, which is -413,000 J.