5. The Free Fall ride is one in which passengers are dropped and experience the

Question

5. The Free Fall ride is one in which passengers are dropped and experience the acceleration of gravity, before being rapidly brought to a complete stop. One such ride, formerly at Magic Mountain, was 13 meters tall. An object falling under the force of gravity has height modeled by the equation H(t) =-4.9t2 + vt + h, where H is measured in meters, u is the initial velocity in meters/second, and h is the initial height in meters. (a) Keeping in mind that the ride drops passengers from a stop, write a function equation to model the height of passengers as a function of time since the passengers started freefall (b) If riders are allowed to free fall for 10 meters before the brakes are applied, how long (in seconds) is the free fall portion of the ride? 65 (c) If the speed of passengers during the freefall portion of the ride is given by the equation S(t)- 9.8t, where S is measured in meters per second. How fast are passengers traveling at the moment when the brakes are first applied? (d) If you want your riders to reach speeds of 30 miles per hour, how long must riders be allowed to free fall to reach this speed? At what height would you apply the brakes? (1 meter/second is the same as 2.24 miles/hour.)

Explanation / Answer

H(t) = -4.9t^2 + vt + 13
because initially, the height, h = 13

And since we start from a stop,
we have v = initial velocity = 0

So,
H(t) = -4.9t^2 + 13

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b)
s = vt + 1/2*gt^2

s = 0 + 4.9 * t^2

10 = 4.9t^2

t^2 = 10/4.9

t^2 = 2.0408163265306122

t = 1.4285714285714285557

So, free fall time is approx 1.43 seconds

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c)
9.8 * 1.4285714285714285557

i.e 13.99999999999999984586

So, approx 14 m/s is the velocity of travel

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d)
30 mph is same as 30/2.24 = 13.3928571428571429 m/s

S = 9.8*t

13.3928571428571429 = 9.8 * t

t = 13.3928571428571429/9.8

t = 1.3666180758017492755 seconds

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