A jet plane is flying with a constant speed along a straight line, at an angle of 25.0° above the horizontal, as Figure 4.30a indicates. The plane has weight whose magnitude is 86500 N, and its engines provide a forward thrust of 103000 N. In addition, the lift force (directed perpendicular to the wings) is 78400 N and the air resistance is 66400 N. Suppose that the pilot suddenly jettisons 2800 N of fuel. If the plane is to continue moving with the same velocity under the influence of the same air resistance , by how much does the pilot have to reduce the thrusst and the lift?
To find out how much the pilot needs to reduce the thrust and the lift in order to maintain the same velocity, we need to analyze the forces acting on the plane before and after jettisoning the fuel.
Let's break down the forces acting on the plane:
1. Weight (W): The weight of the plane acts vertically downwards with a magnitude of 86500 N.
2. Thrust (T): The forward thrust of the engines acts in the direction of motion of the plane. Initially, its magnitude is 103000 N.
3. Lift (L): The lift force acts perpendicular to the wings. Initially, its magnitude is 78400 N.
4. Air Resistance (R): The air resistance opposes the motion of the plane. Initially, its magnitude is 66400 N.
Given that the plane is flying at a constant velocity and angle, we know that the sum of forces in the horizontal direction and the sum of forces in the vertical direction must be zero.
Before jettisoning the fuel:
Sum of horizontal forces:
T - R = 0
Sum of vertical forces:
L + W = 0
After jettisoning the fuel:
The weight remains the same, but the thrust and lift must be reduced.
Let's assume the pilot reduces the thrust by x and the lift by y.
Sum of horizontal forces:
(T - x) - R = 0
Sum of vertical forces:
(L - y) + W = 0
Since the plane is to continue moving with the same velocity, the sum of horizontal forces and the sum of vertical forces must still be zero.
Solving the equations:
(T - x) - R = 0
(L - y) + W = 0
Substituting the given values:
(103000 - x) - 66400 = 0
(78400 - y) + 86500 = 0
Simplifying the equations:
x = 103000 - 66400
y = 78400 + 86500
x = 36600 N
y = 164900 N
Therefore, the pilot needs to reduce the thrust by 36600 N and the lift by 164900 N in order to maintain the same velocity under the same air resistance after jettisoning the fuel.