Question

A white billiard ball with mass m_w = 1.33 kg is moving directly to the right with a speed of v = 2.96 m/s and collides elastically with a black billiard ball with the same mass m_b = 1.33 kg that is initially at rest. The two collide elastically and the white ball ends up moving at an angle above the horizontal of ?_w = 29

Answer 1

m_w = mass of white ball = 1.33 kg

m_b = mass of black ball = 1.33 kg

V_wix = velocity of white ball before collision along X-direction = 2.96 m/s

V_wiy = velocity of white ball before collision along Y-direction = 0 m/s

V_bix = velocity of blaack ball before collision along X-direction = 0 m/s

V_biy = velocity of blaack ball before collision along Y-direction = 0 m/s

V_wf = velocity of white ball after collision

V_wfx = velocity of white ball after collision in X-direction = V_wf cos29

V_wfy = velocity of white ball after collision in Y-direction = V_wf Sin29

V_bf = velocity of black ball after collision

V_bfx = velocity of black ball after collision in X-direction = V_bf cos61

V_bfy = velocity of black ball after collision in Y-direction = -V_bf Sin61   (since it is along negative y-direction)

Using conservation of momentum along X-direction ::

m_wV_wix+ m_bV_bix= m_w V_wfx + m_bV_bfx

inserting the values

2.96 + 0 = V_wf cos29 + V_bf cos61

V_wf (0.875) + V_bf (0.485) = 2.96                         eq-1

Using conservation of momentum along Y-direction ::

m_wV_wiy+ m_bV_biy= m_w V_wfy + m_bV_bfy

inserting the values

0 + 0 = V_wf Sin29 -V_bf Sin61

  V_wf Sin29 = V_bf Sin61                    

V_wf = 1.804 V_bf                             eq-2

Using eq-1 and eq-2

(1.804 V_bf ) (0.875) + V_bf (0.485) = 2.96  

V_bf = 1.4345 m/s          --------------------final speed of black ball

V_wf = 1.804 V_bf   = 1.804 x 1.4345

V_wf = 2.588 m/s          ----------------------- final speed of white ball

final total momentum = initial total momentum = m_w V_wi= 1.33 x 2.96 = 3.937 kgm/s

final total energy = (0.5) m_w V²_wf + (0.5) m_b V²_bf = (0.5) (1.33) (2.588² + 1.4345²)

Final total energy = 5.822 J