Two shuffleboard disks of equal mass, one orange and the other green, are involv

Question

Two shuffleboard disks of equal mass, one orange and the other green, are involved in a perfectly elastic glancing collision. The green disk is initially at rest and is struck by the orange disk moving initially to the right at vOi = 5.05 m/s as in Figure a, shown below. After the collision, the orange disk moves in a direction that makes an angle of = 39.0° with the horizontal axis while the green disk makes an angle of = 51.0° with this axis as in Figure b. Determine the speed of each disk after the collision. vof = m/s vgf = m/s

Explanation / Answer

Initial momentum along the x-axis … Px = M • Voi (green disk is at rest )
Initial momentum along the y-axis … Py = 0 ( no component along y axis before collision)   
Final momentum along the x-axis …
… Px = M • Vof • cos + M • Vgf • cos
Final momentum along the y-axis
… Py = M • Vof • sin - M • Vgf • sin (subtraction of green due to opposite motion along y component)
By conservation of momentum along the x-axis (Px = Px)
… M • Voi = M • Vof • cos + M • Vgf • cos
… Voi = Vof • cos + Vgf • cos (since … M = M )
By conservation of momentum along the y-axis (Py = Py)
… 0 = M • Vof • sin - M • Vgf • sin —> Vof • sin = Vgf • sin ............(1)
From conservation of momentum along the x-axis
… Voi - Vof • cos = Vgf • cos .............(2)
Dividing equation 1 by 2 , we get
[ Vof • sin ] / [ Voi - Vof • cos ] = [ Vgf • sin ] / [ Vgf • cos ]
Vof • sin = [ Voi - Vof • cos ] tan = Voi tan - Vof • cos tan
Vof • sin + Vof • cos tan = Vof • [ sin + cos tan ] = Voi tan
Vof = Voi tan / [ sin + cos tan ]
= ( 5.05 m / s ) tan 51° / [ sin 39° + cos 39° tan 51° ] = 3.92 m / s ............Ans.
Now From equation 1
… Vgf = Vof • sin / sin = ( 3.92 m / s ) sin 39° / sin 51° = 3.18 m / s ...........Ans.

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