A small roller coaster has a height H and a big rollercoaster is 9 times as high with a height of 9H. At the BOTTOM of the rollercoaster, the speed of the bigger rollercoaster is how much faster than the smaller one?

speed at the bottom of an roller coaster depends on the work done by gravity which in turn it depends on the height of that roller coaster

So for the first roller coaster we can write using the conservation of mechanical energy, namely potential + kinetic:

1/2 * m*v1^2=m*g*H_1

where m is the mass of a person on that roller coaster, v_1 is its speed at the bottom of the roller coaster, and g is the gravity.

From that formula we get, after a simple simplification:

v_1=sqrt(2*g*H_1)

Now, if we apply the first formula to the second roller coaster, we get:

v_2=sqrt(2*g*H_2)=sqrt(2*g*9*H_1)=3*sqrt(2*h*H_1)=3*v_1

In other words the second roller coaster is three times faster than the first roller coaster

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