Question

an 8.49 kg bird soars 88.1 m above the ground. a headwind keeps the bird stationary. it then closes its wings and dives down to a tree branch 21.0 m off the ground. assuming negligible air resistance, what is the speed of the bird when it reaches the branch?

38.9 m/s
36.3 m/s
33.3 m/s
35.1 m/s

Explanation:

Identify the given physical parameters

We first extract the given values from the problem statement:

• Mass of the bird, \(m = 8.49\text{ kg}\)
• Initial height, \(h_i = 88.1\text{ m}\)
• Final height, \(h_f = 21.0\text{ m}\)
• Initial speed, \(v_i = 0\text{ m/s}\) (since the bird is stationary before diving)
• Acceleration due to gravity, \(g = 9.8\text{ m/s}^2\)

Calculate the vertical displacement

The bird dives vertically downward from its initial height to the tree branch. We calculate the vertical distance traveled, \(d\):

$$ d = h_i - h_f = 88.1\text{ m} - 21.0\text{ m} = 67.1\text{ m} $$

Apply the kinematic equation for free fall

Since air resistance is negligible, the bird is in free fall. We use the kinematic equation relating final velocity, initial velocity, acceleration, and displacement:

$$ v_f^2 = v_i^2 + 2gd $$

Substituting \(v_i = 0\):

$$ v_f^2 = 2gd $$

Calculate the final speed of the bird

We substitute the known values into the equation to find the final speed \(v_f\):

$$ v_f = \sqrt{2 \cdot 9.8\text{ m/s}^2 \cdot 67.1\text{ m}} $$

$$ v_f = \sqrt{1315.16} \approx 36.265\text{ m/s} $$

Rounding to three significant figures gives \(36.3\text{ m/s}\).

Answer:

• 38.9 m/s
• 36.3 m/s (Correct answer)
• 33.3 m/s
• 35.1 m/s