The Cycloid Ride

THE CYCLOID RIDE


This is the graph for the parametric equation, the graph shows the path a rider will fillow on the cycloid ride.

X= r t - (sin t r)
Y= r – (cos t r)


The y-axisrepresents the distance the rider is above, under or in the horizontal ramp in which the inner circle rolls.(The rider is positioned at 1 metter below the horizontal ramp).
The x-axis represents thedistance the rider is before or after the starting position.
Information given:
-One revolution of the ride takes 90 seconds to complete, the whole ride lasts for 2 complete revolutions, so thecomplete ride lasts 180 seconds.
One complete revolution= 360°, 90 seconds.
So we can find the angular velocity of the ride, and that´ll help us to find the amount of seconds the rider travelsbackwards.
We know that the starting position is (0,-1) and the rider moves backwards to y=0 so we can use the parametric equation to now the degrees the circle moved so that the rider was at y= 0.
Weonly know that y=0 so we can subtitute it in the parametric equation to find the number of degrees the ride moved.
Y= r – (cos t r)
0= 5 – (cos t 7)
(cos t 7)=5
cos t = 5/7
t=cos(-1) (5/7)
t=44.4Therefore it takes 11.1 seconds to the ride to go from y=-1 to y=0 so the first time interval is 0 < t < 11.1
It takes 90 seconds to complete the revolution.
The second time intervalwhere the ride is moving backwards is.
90- 11.1 seconds < t < 90+11.1 seconds
78.9 < t < 101.1
To find the third and final interval we know that the 2 revolutions take 180 seconds to becompleted therefore the 3rd time interval is
180-11.1= 168.9 seconds
168.9 < t < 180
So all the time intervals in which the rider will be moving backwards are.
-0 seconds < t < 11.1secconds (11 seconds)
-90- 11.1 seconds < t < 90+11.1 seconds (22 seconds)
-168.9 seconds < t < 180 seconds (11 seconds)

The rider will be moving backwards for a total of 44 seconds.