Information: When traveling at top speed, a roller coaster train with a mass of 12,000 kg has a velocity of 30 m/s. The kinetic energy of the train at top speed is 5,400,000 J. Math Proof:

KE = 1/2mv2

KE = 1/2(12,000kg)(30m/s)2

KE = 5,400,000J

5,400,000J = 1/2(12,000kg)(30m/s)2

More information: We can assume the KE and PE have an inverse relationship in this system. (Relationship in math and science is not love/Valentines. It is cause and effect; one thing affects another and vice-a-versa. Inverse means when two variables do the opposite. (E. g. When one goes up, the other goes down. When the other goes up, then the first goes down.)

Therefore PE = 5,400,000J, (same as the KE), but at an opposite point on the roller coaster track

Bonus challenge:

Using this formula and the math figures from above, can you calculate how tall the tallest hill was for this roller coaster ride? Please state your answer in meters.

PE = mgh

h=PE/(mg)

h=PE/(12,000kg x 9.8m/s2 )

h=

Lake happy times contains 100 km3 of pure water. it is fed by a river at a rate of 50 km3/yr. at time zero, there is a factory on one shore of lake happy times that begins introducing a pollutant to the lake at a rate of 2km3/yr. there is another river that is fed by lake happy times at a rate that keeps the volume of lake happy times constant. this means that the rate of flow from lake happy times into the outlet river is 52km3/yr. in turn, the flow from this outlet river goes into another lake, lake sad times, at an equal rate. finally, there is an outlet river from lake sad times flowing at a rate that keeps the volume of lake sad times at a constant 100 km3 . (a) find the amount of pollutant in lake sad times at the end of 3 months. (b) at the end of 3 months, observers close the factory due to environmental concerns and no further pollu- tant enters lake happy times. how long will it take for the pollutant in lake sad times (found in part (a)) to be cut in half?