Question

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).

Short answer

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

Step-by-step solution

Vertical force: F_v = mg = (2800 kg)(9.81 m/s^2) F_v = 27468 N (approx.) #Step 2: Calculate the horizontal force on the airplane.# We can also use Newton's second law of motion to calculate the horizontal force on the airplane.

Horizontal force: F_h = ma = (2800 kg)(0.86 m/s^2) F_h = 2408 N (approx.) #Step 3: Calculate the angle.# Now, using the calculated horizontal and vertical forces, we can calculate the angle of the air force.

Angle theta: tan(θ) = F_h / F_v θ = arctan(F_h / F_v)

b) Calculate the new acceleration components.# #Step 4: Calculate the magnitude of force exerted by the air.# Using the Pythagorean theorem, we find the magnitude of force exerted by the air.

Magnitude of air force: F_air = √(F_v^2 + F_h^2) F_air = √(27468^2 + 2408^2) = 27560 N (approx.) #Step 5: Calculate the new angle.# Now, we subtract 2.0° from the initial angle to find the new angle of the air force.

New angle: θ_new = θ - 2.0° #Step 6: Calculate the new horizontal and vertical components of the acceleration.# We can find the new horizontal and vertical components of the acceleration using the new angle and the magnitude of force exerted by the air.

New horizontal component of acceleration (a_h_new): a_h_new = (F_air * cos(θ_new)) / m

New vertical component of acceleration (a_v_new): a_v_new = (F_air * sin(θ_new)) / m