15. An amusement park ride consists of a cart with some riders (of total mass 20

Question

15. An amusement park ride consists of a cart with some riders (of total mass 200 kg) that is set in motion by a large
spring with spring constant 800 N/m. The cart travels along the flat horizontal section of track that is located 17 m
above the ground and then down a ramp toward the loop-the-loop which starts at the ground and has an unknown
diameter. The entire track is frictionless.
a. If the spring is initially compressed by 4 m, what is the speed of the cart as it leaves the spring? 8 m/s, 16 mph
seems like a reasonable roller coaster speed
b. What is the speed of the cart at the bottom of the ramp before reaching the loop-the-loop? 20 m/s, 40 mph seems
fast enough to get some screams
c. What is the maximum height of the loop-the-loop that the cart can do without falling? 16 m, which is about the
height of 8 people standing on top of each other

Explanation / Answer

when the cart is at the maximum height the velocity cannot be zero,as the velocity must compensate for the gravitational pull towards the ground.at the top height the weight is downwards and the force due to velocity which is the centrifugal force acting outwards which is mass*(velocity^2)/radius

M*g=M*V*V/R

R=V*V/g

now the velocity at top will be calculated by using conservation of energy

at bottom the velocity of cart is 20m/s

0.5*M*20*20=0.5*M*V*V+M*g*(2*R)

substituting V*V=g*R

0.5*M*20*20=0.5*M*g*R+2*M*g*R

2.5*g*R=0.5*20*20

R=8.15m

therefore maximum height can be 2*R which is16m

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