At a city park, a person throws some bread into a duck pond. Two 4.00-kg ducks and a 9.00-kg goose paddle rapidly toward the bread, as shown in our sketch (Intro 1 figure) . If the ducks swim at 1.10 m/s, and the goose swims with a speed of 1.30 m/s, the total momentum of the three birds is 8.52 kg\ m/s at 58.9 above the positive x axis.

Find the total momentum of the birds if the goose reverses direction

below the positive x axis

To find the total momentum of the birds if the goose reverses direction below the positive x-axis, we need to take into account the change in direction of the goose's velocity vector.

Given that the total momentum of the three birds in the original scenario is 8.52 kg m/s at 58.9 degrees above the positive x-axis, we can break down the momentum into its x and y components.

The x-component of the momentum can be calculated using the formula:
Px = mvx = (m1v1x + m2v2x + m3v3x)
Where m1, m2, and m3 are the masses of the ducks and goose, and v1x, v2x, and v3x are the horizontal components of their velocities.

Similarly, the y-component of the momentum can be calculated using the formula:
Py = mvy = (m1v1y + m2v2y + m3v3y)
Where v1y, v2y, and v3y are the vertical components of their velocities.

Since the ducks are moving at 1.10 m/s, their velocities can be written as:
v1x = 1.10 cos(58.9°) = 0.57 m/s
v1y = 1.10 sin(58.9°) = 0.93 m/s

And since the goose is moving at 1.30 m/s, its velocity can be written as:
v3x = -1.30 cos(58.9°) = -0.67 m/s (reversed direction)
v3y = -1.30 sin(58.9°) = -1.07 m/s (reversed direction)

Now we can calculate the x-component of the total momentum:
Px = (m1 * v1x) + (m2 * v2x) + (m3 * v3x)
= (4.00 kg * 0.57 m/s) + (4.00 kg * 0.57 m/s) + (9.00 kg * -0.67 m/s)
= 2.28 kg m/s + 2.28 kg m/s - 6.03 kg m/s
= -1.47 kg m/s

And the y-component of the total momentum:
Py = (m1 * v1y) + (m2 * v2y) + (m3 * v3y)
= (4.00 kg * 0.93 m/s) + (4.00 kg * 0.93 m/s) + (9.00 kg * -1.07 m/s)
= 3.72 kg m/s + 3.72 kg m/s - 9.63 kg m/s
= -2.19 kg m/s

Finally, the total momentum of the three birds when the goose reverses direction below the positive x-axis is given by the magnitude of the vector formed by the x and y components:
Total momentum = √(Px² + Py²)
= √((-1.47 kg m/s)² + (-2.19 kg m/s)²)
= √(2.16 + 4.80) kg m/s
= √6.96 kg m/s
= 2.64 kg m/s (rounded to two decimal places)

Therefore, the total momentum of the birds, when the goose reverses direction below the positive x-axis, is approximately 2.64 kg m/s.