Physics for Scientists and Engineers: Foundations and Connections
Physics for Scientists and Engineers: Foundations and Connections
1st Edition
ISBN: 9781337026345
Author: Katz
Publisher: Cengage
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Chapter 10, Problem 30PQ

A billiard player sends the cue ball toward a group of three balls that are initially at rest and in contact with one another. After the cue ball strikes the group, the four balls scatter, each traveling in a different direction with different speeds as shown in Figure P10.30. If each ball has the same mass, 0.16 kg, determine the total momentum of the system consisting of the four balls immediately after the collision.

Chapter 10, Problem 30PQ, A billiard player sends the cue ball toward a group of three balls that are initially at rest and in

FIGURE P10.30

Expert Solution & Answer
Check Mark
To determine

Total momentum of four balls after the collision.

Answer to Problem 30PQ

The total momentum of the four balls after collision is (0.010i^0.028j^)_.

Explanation of Solution

The player sends a cue ball to hit on a group of four balls. All the four balls were initially at rest and after their collision with the cue ball each of them scattered to different directions.

Write the equation to find the total momentum of the four balls.

  Ptot=m1v1+m3v3+m6v6                                                                                   (I)

Here,Ptot is the total momentum of four balls,m1 is the mass of first ball,m3 is the mass of ball 3, m6 is the mass of ball 6, v1 is the velocity of ball 1, v3 is the velocity of ball 3 and v6 is the velocity of ball 6.

Refer figure 10.30 provided in the problem. The figure gives the coordinate system of the system of four balls.

Consider each ball and write the component form of velocity vector for each ball.

Write the equation to find the velocity vector for ball 1.

  v1=v1xcosθi^+v1ysinθj^                                                                                   (II)

Here, v1x is the x component of velocity vector of ball 1,v1y is the y component of velocity vector of ball 1,θ is the angle made by vector and the x axis,i^ is the unit vector along x axis, and j^ is the unit vector along y axis.

Write the equation to find the velocity vector for ball 3.

  v3=v3xcosθi^+v3ysinθj^                                                                                (III)

Here, v3x is the x component of velocity vector of ball 3 and v3y is the y component of velocity vector of ball 3.

Write the equation to find the velocity vector for ball 6.

  v6=v6xcosθi^+v6ysinθj^                                                                                (IV)

Here, v6x is the x component of velocity vector of ball 6,v6y is the y component of velocity vector of ball 6

Write the equation to find the velocity vector for cue ball.

  vc=vxcosθi^+vysinθj^                                                                                   (V)

Conclusion:

Substitute 0.35m/s for vx,0.35m/s for vy and 60° for θ in equation (II) to get v1.

  v1=(0.35m/s)cos(60°)i^+(0.35m/s)sin(60°)j^=(0.175i^+0.303j^)m/s

Substitute 0.45m/s for vx,0.45m/s for vy and 60° for θ in equation (III) to get v3.

  v3=(0.45m/s)cos(60°)i^(0.45m/s)sin(60°)j^=(0.390i^0.225j^)m/s

Substitute 0.25m/s for vy and 90° for θ in equation (IV) to get v6.

  v6=(0.25m/s)sin(90°)j^=(0.25j^)m/s

Substitute 0.50m/s for vx and 0° for θ in equation (V) to get vc.

  vc=(0.50m/s)cos(0°)i^=(0.50i^)m/s

Substitute 0.16kg for m1,m2,m3andmc, (0.175i^+0.303j^)m/s for v1, (0.390i^0.225j^)m/s for v3, (0.25j^)m/s for v6 and (0.50i^)m/s for vc in equation (I) to get Ptot.

  Ptot=(0.16kg)[(0.175i^+0.303j^)m/s+(0.390i^0.225j^)m/s+(0.25j^)m/s+(0.50i^)m/s]=(0.028i^+0.048j^)kgm/s+(0.062i^0.036j^)kgm/s0.040j^kgm/s0.080i^kgm/s=(0.010i^0.028j^)

Therefore, the total momentum of the four balls after collision is (0.010i^0.028j^)_.

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Chapter 10 Solutions

Physics for Scientists and Engineers: Foundations and Connections

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