At what velocity does the stone impact? (click on link for picture)?

Question

I have no idea how to do this problem, so if you can walk me through step by step, it would be very much appreciated!

http://i231.photobucket.com/albums/ee10/mitochondria3/DavidGoliath.jpg

Answer 1

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Radius of circle=r=1m

frequency=f=3rev/s

angular velocity=w=2(pi)f=6(pi) radian/second

Initial velocity of stone=u=rw=1*6(pi) =6*3.1415=18.8495 m/s

The initial velocity makesO= 30 degree with horizontal

horizontal component of initial velocity=ucosO=18.8495*0.867=16.3425 m/s

horizontal component of velocity remains constant and equal to 16.3425 m/s

horizontal displacement =x=10 m

time taken to cover x=t=10 / 16.3425=o.6119 s

vertical component of initial velocity=usinO=18.8495*0.5=9.4248 m/s

acceleration in vertical direction=acceleration of gravity=g=9.8m/s^2

vertical displacement=y=5m

vertical component of final velocity=v sin theta=usin30 -gt

vertical component of final velocity =v sin theta =9.4248 -9.8*0.6119

vertical component of final velocity = v sin theta=3.4282 m/s

horizontal component of final velocity =v cos theta= 16.3425 m/s

final velocity=v= sq rt [ (v sin theta)^2 + (v cos theta)^2 ]


v = sq rt [ (3.4282 )^2 + (16.3425 )^2 ]=16.6981 m/s

final velocity=v=16.6981 m/s

tan theta =3.4262 / 16.3425=0.20964

theta =11.14 degree

The stone impacts with velocity 16.6981 m/s at angle11.14 degree with horizontal

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Answer 2

Before release, the stone is tracing out a circle of circumference 2π meters, three times a second. So the velocity of the stone at release is 6π meters per second (m/s), or about 18.85 m/s, at an angle of 30° with the horizontal. So at release, the horizontal component of velocity is 6π cos 30° = (3π√3) m/s (about 16.32 m/s) and the vertical component of velocity is 6π sin 30° = 3π m/s (about 9.42 m/s).

Ignoring air resistance, the horizontal component remains 3π√3 m/s. The time it takes, in seconds, for the stone to travel 10 meters horizontally is 10/(3π√3) (or about 0.6 sec) This is the moment of impact.

During its flight the stone is acted on vertically by gravity, so the vertical component of velocity at time t (in seconds) is 3π - gt = 3π - 9.8t

Evaluate this at the time of impact to get the vertical component of velocity at that time.

The magnitude of the velocity at impact is the square root of the sum of the squares of the horizontal and vertical components of velocity at the moment of impact. As you should be able to tell from a diagram, the tangent of the angle that the velocity makes with the horizontal is

(vertical component of velocity at impact)/(horizontal component of velocity at impact)