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 find the direction of the force exerted on the airplane by the air, we need to first understand the forces acting on the airplane.

The force exerted on the airplane can be resolved into two components: the vertical component and the horizontal component. The vertical component is responsible for the plane's altitude change, while the horizontal component is responsible for the change in its horizontal velocity.

(a) To find the direction of the force exerted on the airplane by the air, we need to consider the forces acting in the vertical direction. The presence of a constant vertical velocity indicates that the net vertical force acting on the airplane is zero. This means that the force exerted by the air upward is equal in magnitude to the force of gravity acting downward.

We can calculate the force of gravity using the formula F = mg, where F is the force, m is the mass, and g is the acceleration due to gravity.

Fgravity = mg
Fgravity = 2800 kg * 9.81 m/s^2
Fgravity ≈ 27468 N (downward)

Since the upward force exerted by the air must be equal in magnitude and opposite in direction to the force of gravity, the direction of the force exerted by the air on the airplane is downward (opposite to the plane's ascending direction).

(b) To find the horizontal and vertical components of the plane's acceleration, we can use the formula a = F/m, where a is acceleration, F is force, and m is mass.

Given that the magnitude of the force due to air is the same as before but with a direction 2.0° closer to the vertical, we need to find the new components of the force.

Let's call the original direction of the force F1 and the new direction F2.

The horizontal component of acceleration (ah) can be calculated using:

ah = F2h / m

The vertical component of acceleration (av) can be calculated using:

av = F2v / m

To find F2h and F2v, we need to decompose the force F2 into its horizontal and vertical components.

F2h = F2 * cosθ
F2v = F2 * sinθ

θ is the angle between the force and the horizontal direction.

Given that F2 has the same magnitude but a direction 2.0° closer to the vertical than F1, we need to calculate θ2 as follows:

θ2 = θ1 - 2.0°

Once we have calculated θ2, we can find F2h and F2v using the equations mentioned earlier.

Then, we substitute the values into the formulas for ah and av:

ah = F2h / m
av = F2v / m

This will give us the horizontal and vertical components of the plane's acceleration.