At a particular instant, a stationary observer on the ground sees a package falling from a moving airplane with a speed vobserver at an angle θ to the vertical. To the pilot flying horizontally at a constant speed relative to the ground the package appears to be falling vertically with a speed vpilot at that same instant. What is the speed of the pilot relative to the ground in terms of the given quantities?
Vpilot = Vobserver sin θ
To find the speed of the pilot relative to the ground, we need to consider the velocities involved and use vector addition.
Let's break down the velocities involved:
1. Velocity of the package relative to the observer (on the ground):
This velocity can be broken down into two components:
- Vertical component: v_vertical = v_observer * sin(θ)
- Horizontal component: v_horizontal = 0 (since the package falls vertically)
2. Velocity of the package relative to the pilot (on the airplane):
- Vertical component: v_vertical_pilot = vpilot
- Horizontal component: v_horizontal_pilot = 0 (since the package falls vertically)
3. Velocity of the observer (on the ground) relative to the ground:
- Vertical component: v_vertical_observer = 0 (since the observer is stationary vertically)
- Horizontal component: v_horizontal_observer = 0 (since the observer is stationary horizontally)
4. Velocity of the pilot (on the airplane) relative to the ground: v_pilot_ground
Using vector addition, we can calculate the velocity of the package relative to the ground:
v_relative_ground = v_relative_pilot + v_relative_observer
Since the vertical components of both the pilot and observer are zero, we can ignore them:
v_horizontal_pilot + 0 = v_horizontal_package + 0
Now, we can see that the horizontal components of the package velocity and the relative velocities cancel each other out. Therefore, we can conclude that:
v_pilot_ground = 0
The speed of the pilot relative to the ground is zero.