Plssss help, sorry for the trouble

Question

Link to image of the problem

Answer 1

This is just a problem in conservation of energy as a function of height (assuming no friction in the coaster wheels or air resistance!!!).

For total energy as a sum of kinetic and potential energies, we can write

TE = KE + GPE = 0.5*m*v² + m*g*h (1)

where m = gross coaster mass = 1500 kg

v = roller coaster velocity at some point in m/s

h = roller coaster height at some point in meters

g = 9.81 m/s²

At point A, we can express the total energy as

TE = 0.5*m*v(A)² + m*g*h(A) (2)

And similarly, since total energy is conserved, for any point X

TE = 0.5*m*v(X)² + m*g*h(X) (3)

Combining (2) and (3), and solving for the unknown velocity v(X),

0.5*m*v(A)² + m*g*h(A) = 0.5*m*v(X)² + m*g*h(X)

m*g*h(A) - m*g*h(X) = 0.5*m*v(X)² - 0.5*m*v(A)²

m*g* [ h(A) - h(X) ] = 0.5*m* [ v(X)² - v(A)² ]

v(X)² - v(A)² = 2*g* [ h(A) - h(X) ]

v(X)² = v(A)² + 2*g* [ h(A) - h(X) ]

v(X) = sqrt{ v(A)² + 2*g* [ h(A) - h(X) ] }

Putting in the given data for v(A) and h(A), we have

v(X) = sqrt{ (2)² + 2*9.81* (20 - h(X) } = sqrt{4 + 19.62*(20 - h(X) }

At point C, the height is half of D, or h(C) = 6 m. Therefore,

v(C) = sqrt{ 4 + 19.62*(20-6) } = 16.7 m/s

At point E, h(E) = 10 m, so that

v(E) = sqrt{ 4 + 19.62*(20-10) } = 14.1 m/s