An airplane of mass \(2800 \mathrm{kg}\) has just lifted off the runway. It is gaining altitude at a constant \(2.3 \mathrm{m} / \mathrm{s}\) while the horizontal component of its velocity is increasing at a rate of $0.86 \mathrm{m} / \mathrm{s}^{2} .\( Assume \)g=9.81 \mathrm{m} / \mathrm{s}^{2} .$ (a) Find the direction of the force exerted on the airplane by the air. (b) Find the horizontal and vertical components of the plane's acceleration if the force due to the air has the same magnitude but has a direction \(2.0^{\circ}\) closer to the vertical than its direction in part (a).

Answer: To find the new horizontal and vertical components of the airplane's acceleration, follow these steps: 1. Calculate the vertical and horizontal forces acting on the airplane using Newton's second law of motion (F = ma) and the given mass and accelerations. 2. Calculate the angle of the air force using the arctan function and the calculated vertical and horizontal forces. 3. Subtract 2.0° from the initial angle to find the new angle of the air force. 4. Calculate the magnitude of the air force using the Pythagorean theorem and the vertical and horizontal forces. 5. Use the new angle and the magnitude of the air force to find the new horizontal (a_h_new) and vertical (a_v_new) components of the acceleration by dividing the forces by the mass of the airplane: - a_h_new = (F_air * cos(θ_new)) / m - a_v_new = (F_air * sin(θ_new)) / m