Billiard Balls Collision at an Angle

Introduction

• Discussion about collision of billiard balls at an angle
• One ball is moving at 5 m/s, and the other is at rest
• After collision, first ball moves at a 40° angle from its original direction
• The second ball moves in a different direction
• Goal: Determine the speeds of the balls after collision

Key Concepts

Initial Conditions

• V_1i: Initial velocity of the first ball = 5 m/s
• V_2i: Initial velocity of the second ball = 0 m/s (at rest)
• Collision Angle: First ball moves at a 40° angle post-collision

Conditions for 90° Angle of Separation

• Elastic collision
• Equal mass of the two balls

Angle Calculations

• Angle of separation is 90° if both conditions are met
• Given angle for the first ball = 40°
• Calculated angle for the second ball = 50°

Conservation of Momentum

• Momentum is conserved along both x-axis and y-axis

X-Axis Momentum

• Initial momentum: ( M , V_1i )
• Post-collision, resolve velocities into components:
  • First ball: ( M , V_{1f} , \cos(\theta) )
  • Second ball: ( M , V_{2f} , \cos(\alpha) )
• X-axis conservation equation: [ M , V_1i = M , V_{1f} , \cos(\theta) + M , V_{2f} , \cos(\alpha) ]
• Mass cancels out, angle values plugged in:
  • ( \cos(40°) = 0.77 )
  • ( \cos(50°) = 0.64 )

Y-Axis Momentum

• Initial momentum along y-axis: 0 (no initial y-component)
• Post-collision, resolve y-components:
  • First ball: ( M , V_{1f} , \sin(\theta) )
  • Second ball: ( M , V_{2f} , \sin(\alpha) ) (negative direction)
• Y-axis conservation equation: [ M , V_{1f} , \sin(\theta) = M , V_{2f} , \sin(\alpha) ]
• Mass cancels out, solve for relationship:
  • ( V_{1f} = V_{2f} , (1.19) )

Solution

• Use relationship and solve equations:
  • Substitute ( V_{1f} ) in x-axis equation
  • Solve for ( V_{2f} ): 3.21 m/s
  • Substitute ( V_{2f} ) back to find ( V_{1f} ): 3.82 m/s

Conclusion

• Speeds after collision:
  • First ball: 3.82 m/s
  • Second ball: 3.21 m/s
• Illustration of solving using conservation of momentum principles

Additional Notes

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