A plane is flying horizontally with speed 244 m/s at a height 5670 m above the ground, when a package is thrown downward from the plane with a vertical speed v(initial) = 73 m/s.

What horizontal distance is traveled by this package?

The acceleration of gravity is 9.8 m/s2 .
Neglecting air resistance, when the package hits the ground

The time to reach the ground from an altitude of 5670 meters derives from h = Vo(t) + g(t^2)/2 where Vo = 73m/s.

Therefore, 5670 = 73t + 4.9t^2 from which t = 27.373 sec.

The horizontal distance traveled is d = 27.373(244) = 6679m.

To find the horizontal distance traveled by the package, we need to use the equations of motion in both horizontal and vertical directions.

In the horizontal direction, since the plane is flying horizontally with a constant speed, the horizontal velocity of the package will remain constant throughout its motion. Therefore, the horizontal distance traveled by the package will depend on the time it takes to reach the ground.

In the vertical direction, we need to determine the time it takes for the package to hit the ground. We can use the equation of motion in the vertical direction:

h = v(initial) * t + (1/2) * g * t^2

where,
h = height above the ground (5670 m),
v(initial) = initial vertical speed (-73 m/s),
g = acceleration due to gravity (-9.8 m/s^2),
t = time taken to reach the ground.

Rearranging the equation, we get:

-(1/2) * g * t^2 + v(initial) * t - h = 0

This is a quadratic equation in t, which we can solve using the quadratic formula:

t = (-v(initial) +/- sqrt(v(initial)^2 - 4 * (-1/2) * g * (-h))) / (2 * (-1/2) * g)

Substituting the values into the equation, we get:

t = (-73 +/- sqrt(73^2 - 4 * (-1/2) * (-9.8) * (-5670))) / (2 * (-1/2) * (-9.8))

Calculating this equation, we get two values for t: t1 and t2.

Now that we have the time it takes for the package to hit the ground, we can calculate the horizontal distance by multiplying the horizontal velocity of the plane by the time of flight:

horizontal distance = 244 m/s * t1 (or t2, depending on which solution makes sense in this context)

Therefore, to find the horizontal distance traveled by the package, we need to calculate the time it takes to reach the ground using the quadratic equation and then multiply it by the horizontal velocity of the plane.