A roller coaster has a loop-the-loop with a radius of R (diameter of 2R). If the

Question

A roller coaster has a loop-the-loop with a radius of R (diameter of 2R). If the roller coaster is released from some height h, a) How fast does the roller coaster have to be going at the top of the loop, b) How much energy does the roller coaster have at the top of the loop, and c) what must h be to make sure that the roller coaster makes it through the loop? 10. If the kinetic energy of an object is doubled what happens to its speed? What if it the kinetic energy is quadrupled? If an object's speed is doubled, what happens to its kinetic energy? What if the speed is quadrupled?

Explanation / Answer

Here,

a) at the top of the loop ,

let the speed at the top of loop is v

Using conservation of energy

0.5 * m * v^2 = m *g * (h - 2R)

v = sqrt(2 * g * (h - 2R))

the speed of roller coaster at the top of the loop is sqrt(2 * g * (h - 2R))

b)

at the top of loop

energy at the top of roller coaster = m * g * h

c)

for the roller coaster to go round the loop

m * v^2/R = m * g

m * (2 *g * (h - 2R))/R = m * g

2 * (h - 2R) = R

h = 2.5 R

the minimum height needed is 2.5 R

10)

as the kinetic energy is given as

KE = 0.5 * m * v^2

for doubling the kinetic energy

speed = sqrt(2) * time

the speed will become sqrt(2) times

----------

if we double the speed

KE = 2^2 * initial energy

the kinetic energy will become 4 times

---------------

if we quadruple the speed

kinetic energy = 4^2 * intial energy

the kinetic energy wil become 16 times

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