Math  /  Geometry

QuestionA falcon dives toward the ground with a constant velocity of 5.10 m/s5.10 \mathrm{~m} / \mathrm{s} at 57.057.0^{\circ} below the horizontal. The Sun is directly overhead and casts a shadow of the falcon directly below it. What is the speed (in m/s\mathrm{m} / \mathrm{s} ) of its shadow on level ground? \square m/s\mathrm{m} / \mathrm{s} Need Help? Read It

Studdy Solution

STEP 1

1. The falcon's velocity can be decomposed into horizontal and vertical components.
2. The shadow moves with the horizontal component of the falcon's velocity.
3. The angle given is measured from the horizontal.

STEP 2

1. Decompose the falcon's velocity into horizontal and vertical components.
2. Determine the speed of the shadow on the ground.

STEP 3

Identify the components of the falcon's velocity. The velocity vector can be split into horizontal (vxv_x) and vertical (vyv_y) components using trigonometric functions. Given the velocity v=5.10m/sv = 5.10 \, \mathrm{m/s} and angle θ=57.0\theta = 57.0^\circ below the horizontal:
vx=vcos(θ) v_x = v \cdot \cos(\theta) vy=vsin(θ) v_y = v \cdot \sin(\theta)

STEP 4

Calculate the horizontal component vxv_x:
vx=5.10m/scos(57.0) v_x = 5.10 \, \mathrm{m/s} \cdot \cos(57.0^\circ)

STEP 5

Calculate the vertical component vyv_y (though not needed for the shadow's speed, it's part of the decomposition):
vy=5.10m/ssin(57.0) v_y = 5.10 \, \mathrm{m/s} \cdot \sin(57.0^\circ)

STEP 6

The speed of the shadow on the ground is equal to the horizontal component of the falcon's velocity, vxv_x. Thus, we only need the value of vxv_x:
vx=5.10m/scos(57.0) v_x = 5.10 \, \mathrm{m/s} \cdot \cos(57.0^\circ)

STEP 7

Calculate vxv_x using cos(57.0)0.544\cos(57.0^\circ) \approx 0.544:
vx=5.10m/s0.544 v_x = 5.10 \, \mathrm{m/s} \cdot 0.544 vx2.78m/s v_x \approx 2.78 \, \mathrm{m/s}
The speed of the falcon's shadow on level ground is approximately:
2.78m/s \boxed{2.78 \, \mathrm{m/s}}

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