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
To find the horizontal distance traveled by the package, we need to calculate the time it takes for the package to reach the ground.
First, let's find the time it takes for the package to hit the ground. We can use the equation of motion:
h = v(initial)t + (1/2)at^2
Where:
h = height of the package from the ground
v(initial) = initial vertical velocity of the package
a = acceleration due to gravity
t = time
Rearranging the equation, we get:
t^2 + (2v(initial)/a)t - (2h/a) = 0
Here, h = 5670 m, v(initial) = -73 m/s (since the package is thrown downward), and a = -9.8 m/s^2 (taking downward direction as negative).
Plugging in these values, the equation becomes:
t^2 + (2*-73)/-9.8 * t - (2*5670)/-9.8 = 0
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
Where a = 1, b = (2*-73)/-9.8, and c = (2*5670)/-9.8.
Now, we can plug in these values to find t. Taking the positive solution since time cannot be negative:
t = [-(2*-73)/-9.8 + √((2*-73)/-9.8)^2 - 4(1)((2*5670)/-9.8)] / 2(1)
Once we find t, we can calculate the horizontal distance traveled by the package. The horizontal distance traveled can be calculated by multiplying the horizontal speed of the plane (244 m/s) by the time taken by the package to hit the ground.
Distance = horizontal speed × time
Therefore, once we have the value of t, we can calculate the horizontal distance traveled by the package.