A plane is traveling at an altitude of 500meters and drops a bottle the bottle lands 400meters horizontally. how fast was the plane traveling when the bottle was released?

The bottle and the plane travel the same horizontal speed and distance

How long to fall 500 m?
500 = (1/2) 9.8 t^2
solve for t

How far in t seconds?
400 = v t
so
v = 400 / t

To determine the speed of the plane when the bottle was released, we can use the principles of projectile motion. We'll consider that the horizontal and vertical motions are independent of each other.

First, we need to find the time it took for the bottle to fall vertically. We know that the acceleration due to gravity, g, is approximately 9.8 m/s² downward.

Using the kinematic equation: h = (1/2)gt², where h is the vertical displacement (500m) and t is the time, we can solve for t:

500 = (1/2) * 9.8 * t²

Simplifying the equation:

t² = 500 / (4.9)
t² ≈ 102.04
t ≈ √102.04
t ≈ 10.1 seconds (rounded to one decimal place)

Now that we have the time, we can calculate the horizontal speed of the plane when the bottle was released. We know that the horizontal displacement is 400 meters, and the time is 10.1 seconds.

Using the equation: speed = distance / time, we can find the speed of the plane:

speed = 400m / 10.1s ≈ 39.6 m/s

Thus, the plane was traveling at approximately 39.6 m/s when the bottle was released.