Log Ride (object sliding down a circularly curved slope). In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a slope into a pool below as shown. The water in the channel ensures that there is very little friction. B Ax On this particular ride, the slope (the black arc through points A and B) is a circular curve of radius R, centered on point P. The dotted line shows the boat's trajectory. At some point B along the slope, the boat (and the water falling with it) will separate from the track and fall freely as shown. Note that the pond is level with point P. Considering the boat as a particle, assume it starts from rest at point A and slides down the slope without friction. a) Determine the angle psep at which the boat will separate from the track. b) Determine the horizontal distance Ax (from point P) at which the boat strikes the pond surface. c) Determine the impact speed vf and impact angle 0. Hints: Derive a formula giving the maximum speed vmax at which the boat can stay on the track, in terms of the angle p. (Circular kinematics.) Derive a formula for the speed v of the boat as it traverses the circular slope, in terms of the angle p. (Conservation of energy.)
Log Ride (object sliding down a circularly curved slope). In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a slope into a pool below as shown. The water in the channel ensures that there is very little friction. B Ax On this particular ride, the slope (the black arc through points A and B) is a circular curve of radius R, centered on point P. The dotted line shows the boat's trajectory. At some point B along the slope, the boat (and the water falling with it) will separate from the track and fall freely as shown. Note that the pond is level with point P. Considering the boat as a particle, assume it starts from rest at point A and slides down the slope without friction. a) Determine the angle psep at which the boat will separate from the track. b) Determine the horizontal distance Ax (from point P) at which the boat strikes the pond surface. c) Determine the impact speed vf and impact angle 0. Hints: Derive a formula giving the maximum speed vmax at which the boat can stay on the track, in terms of the angle p. (Circular kinematics.) Derive a formula for the speed v of the boat as it traverses the circular slope, in terms of the angle p. (Conservation of energy.)
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Transcribed Image Text:Log Ride (object sliding down a circularly curved slope).
In an amusement park ride, a boat moves slowly in a narrow channel of water. It then passes over a
slope into a pool below as shown. The water in the channel ensures that there is very little friction.
A
B
(R
Ax
On this particular ride, the slope (the black arc through points A and B) is a circular curve of radius R,
centered on point P. The dotted line shows the boat's trajectory. At some point B along the slope, the
boat (and the water falling with it) will separate from the track and fall freely as shown. Note that the
pond is level with point P.
Considering the boat as a particle, assume it starts from rest at point A and slides down the slope
without friction.
a) Determine the angle øsen at which the boat will separate from the track.
b) Determine the horizontal distance Ax (from point P) at which the boat strikes the pond surface.
c) Determine the impact speed vf and impact angle 0.
Hints:
Derive a formula giving the maximum speed vmax at which the boat can stay on the track, in
terms of the angle p. (Circular kinematics.)
Derive a formula for the speed v of the boat as it traverses the circular slope, in terms of the
angle p. (Conservation of energy.)
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