You are on the highest point of a Ferris wheel that rotates clockwise and want t

Question

You are on the highest point of a Ferris wheel that rotates clockwise and want to throw a ball so you can catch it when you get to the lowest point of the Ferris wheel. The Ferris wheel has a radius of 25.0 m and is moving with a constant velocity of 4.50m/s. a) With what velocity should you throw the ball and in what direction? b) would this be possible in real life? You are on the highest point of a Ferris wheel that rotates clockwise and want to throw a ball so you can catch it when you get to the lowest point of the Ferris wheel. The Ferris wheel has a radius of 25.0 m and is moving with a constant velocity of 4.50m/s. a) With what velocity should you throw the ball and in what direction? b) would this be possible in real life?

Explanation / Answer

given, velocity opf ferris wheel at the highest point, vo = 4.5 m/s
radius of ferris wheel, R = 25 m

now let assume that the ball is thrown at a velocity v = v(cos(theta)i + sin(theta)j) form the highest point
where i and j are unit vactors along x and y directions
then net velocity of the ball at the highest point is, v = ([vcos(theta) + 4.5)i + vsin(theta)j )

to catch the ball on the ferris wheel after time t
the coordinates of the ball and the person must be the same

horizontal coordinates
(vcos(theta) + 4.5)t = Rsin(wt)
vcos(theta)t = Rsin(wt) - 4.5t
vertical coordinates
vsin(theta)t - 0.5gt^2 = Rcos(wt)

so v^2 = (Rsin(wt) - 4.5t)^2 + (Rcos(wt) + 0.5gt^2)^2
v^2 = R^2sin^2(wt) + 20.25t^2 - 9Rtsin(wt) + R^2cos^2(wt) + 0.25g^2t^4 + gt^2Rcos(wt)
v^2 = R^2 + 20.25t^2 - 9Rtsin(wt) + 0.25g^2t^4 + gt^2Rcos(wt)

lets assume we want to catch the ball at t = pi/w
then
v^2 = R^2 + 20.25*pi^2/w^2 + 0.25g^2*(pi/w)^4 + g(pi/w)^2*R
now w = v/R = 4.5/25 = 0.18 rad/s

v^2 = 25^2 + 20.25*pi^2/0.18^2 + 0.25g^2*pi^4/0.18^4 + 9.81*pi^2*25/0.18^2
v = 1519.6627 m/s

1519.6627*cos(theta)*pi/0.18 + 4.5*pi/0.18 = 0
1519.6627*cos(theta) + 4.5 = 0
cos(theta) = 90.169 deg

a. so one combination can be v = 1519.6627 m/s at 90.169 deg to the horizontal
b. this comnbination will not be possible in real life as thje speed required is too high to be thrown by human hands

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