An airplane with a speed of 88.0 m/s is climbing upward at an angle of 72.1 ° with respect to the horizontal. When the plane's altitude is 779 m, the pilot releases a package. (a) Calculate the distance along the ground, measured from a point directly beneath the point of release, to where the package hits the earth. (b) Relative to the ground, determine the angle of the velocity vector of the package just before impact.

To answer these questions, we need to break down the problem into its horizontal and vertical components, and then solve for the desired values.

(a) To calculate the distance along the ground, we need to find the horizontal component of the package's motion. Since the airplane is climbing upward at an angle of 72.1°, the vertical component of its velocity is given by:

Vertical component = airplane speed * sin(angle)

Vertical component = 88.0 m/s * sin(72.1°)

Vertical component = 88.0 m/s * 0.9384

Vertical component = 82.6 m/s

Next, we can determine the time it takes for the package to hit the ground. We can do this by using the equation of motion for vertical motion:

Vertical displacement = initial vertical velocity * time + (1/2) * acceleration * time^2

Since the package is dropped, its initial vertical velocity is 0 m/s, and the acceleration due to gravity is -9.8 m/s^2 (negative because it acts downward). The vertical displacement is the altitude of the plane, which is 779 m.

779 m = 0 * t + (1/2) * (-9.8 m/s^2) * t^2

Rearranging the equation, we get:

4.9 t^2 - 779 = 0

Solving this quadratic equation will give us the time it takes for the package to hit the ground. Once we have the time, we can calculate the horizontal distance traveled by the package using the formula:

Horizontal distance = horizontal velocity * time

(b) To determine the angle of the velocity vector of the package just before impact, we need to find the resultant velocity of the package in both horizontal and vertical directions. We can do this by using the Pythagorean theorem:

Resultant velocity = sqrt(horizontal velocity^2 + vertical velocity^2)

Once we have the resultant velocity, we can find the angle by using the inverse tangent function:

Angle = atan(vertical velocity / horizontal velocity)

By following these steps, you should be able to calculate both the distance along the ground and the angle of the velocity vector of the package just before impact.