Question

Kingda ka at six flags has a 55m drop. The rollercoaster has a mass of 8000kg. How fast it be travelling when it reaches the bottom?

Answer 1

`32.8\ m/s`

Explanation:

We know from Galileo that mass doesn't effect how fast things fall to Earth. There's the famous experiment of dropping a feather and a hammer and both should fall to the ground at the same time (if there isn't vacuum). This was done by Apollo 15:

as seen here.

Therefore, we only use constant gravitational acceleration. The mass is only given to you as a distraction. This means we can use familiar formulae for constant acceleration.

The most useful one is
`v^2 =v_0^2 - 2a Delta x`
which can be proven by the fact that

`v - v_0 = a t rightarrow t = (v-v_0)/a`
`Delta x = 1/2 at^2 + v_0 t = (v-v_0)^2/(2a) + (v_0(v-v_0))/a`
`2aDelta x = (v-v_0)^2 + 2v_0(v-v_0) = (v-v_0)(v+v_0) = v^2 - v_0^2`

So now we can use this formula for Kingda Ka. You start at rest (meaning `v_0 = 0`) and your acceleration is `g approx 9.8 m/s^2` so

`v^2 = cancel{v_0^2} - 2aDeltax = 2 * g * (55\ m) approx 1078 m^2/s^2`
`v = 32.8 m/s`

Sidenote: I don't know if you've learned about energy yet, but energy also makes this argument pretty simple. We equate our initial and final energies and get the exact same formula:
`E_0 = mgh and E_f = 1/2 mv^2`
`mgh = 1/2 mv^2 rightarrow 2gh = v^2 xrightarrow 2aDeltax = v^2`