An airplane of mass 2800 Kg has just lifted off the runway. It is gaining altitude at a constant 2.3 M/s while the horizontal component of its velocity is increasing at a rate of 0.86 M/s2. Assume g= 9.81 M/s2. (a) Find the direction of the force exerted on the airplane by the air. (b) Find the horizontal and vertical components of the planes acceleration if the force due to the air has the same magnitude but has a direction of 2.0ᴼ closer to the vertical than its description in part (a).

To answer these questions, we need to analyze the forces acting on the airplane using Newton's second law of motion. Let's break it down step by step.

(a) Find the direction of the force exerted on the airplane by the air:
We know that the airplane is gaining altitude at a constant rate. Therefore, the net force acting on the airplane must be in the upward direction to counteract the force of gravity. In other words, the force exerted on the airplane by the air must have both vertical and horizontal components.

The vertical component of the force, F_vert, can be determined using Newton's second law:

F_vert = m * a_vert

where
m = mass of the airplane (2800 kg)
a_vert = vertical component of the acceleration (g = 9.81 m/s^2)

Substituting the values, we get:

F_vert = 2800 kg * 9.81 m/s^2
F_vert = 27468 N (upward)

The horizontal component of the force, F_horiz, can be determined using the given rate of change of the horizontal velocity:

F_horiz = m * a_horiz

where
a_horiz = rate of change of the horizontal velocity (0.86 m/s^2)

Substituting the values, we get:

F_horiz = 2800 kg * 0.86 m/s^2
F_horiz = 2408 N

Therefore, the force exerted on the airplane by the air has a vertical component of 27468 N upward and a horizontal component of 2408 N.

(b) To find the horizontal and vertical components of the plane's acceleration, we need to consider the new direction of the force due to the air.

Let's say the original direction of the force (in part a) makes an angle θ1 with the vertical. The force in part (b) makes an angle 2° closer to the vertical, which means the new angle is θ2 = θ1 - 2°.

The force due to the air in part (b) has the same magnitude as in part (a), so the vertical and horizontal components of acceleration can be determined using the trigonometric relationships:

a_vert = F_vert / m
a_horiz = F_horiz / m

Substituting the values, we get:

a_vert = 27468 N / 2800 kg
a_horiz = 2408 N / 2800 kg

Calculating these values will give you the horizontal and vertical components of the airplane's acceleration in part (b).

Note: It's important to convert the angle from degrees to radians if your calculations require it.