A plane designed for vertical takeoff has a mass of 9.0 103 kg. Find the net work done by all forces on the plane as it accelerates upward at 11.3 m/s2 through a distance of 34.8 m after starting from rest.
To find the net work done on the plane, we can use the work-energy principle. The work done by all forces can be calculated by finding the change in kinetic energy of the plane.
The initial kinetic energy, K1, of the plane is zero since it starts from rest.
The final kinetic energy, K2, of the plane can be calculated using the equation: K2 = (1/2)mv^2, where m = mass of the plane and v = final velocity of the plane.
Given:
Mass of the plane (m) = 9.0 × 10^3 kg
Acceleration (a) = 11.3 m/s^2
Distance (d) = 34.8 m
First, we need to find the final velocity of the plane.
Using the equation: v^2 = u^2 + 2ad, where u = initial velocity (which is 0 since the plane starts from rest):
v^2 = 0^2 + 2 × 11.3 m/s^2 × 34.8 m
v^2 = 0 + 2 × 11.3 m/s^2 × 34.8 m
v^2 = 2 × 11.3 m/s^2 × 34.8 m
v^2 = 765.12 m^2/s^2
Taking the square root on both sides:
v = √(765.12 m^2/s^2)
v ≈ 27.67 m/s
Now we can calculate the final kinetic energy, K2, using the equation: K2 = (1/2)mv^2:
K2 = (1/2) × 9.0 × 10^3 kg × (27.67 m/s)^2
K2 = (1/2) × 9.0 × 10^3 kg × 767.4889 m^2/s^2
K2 ≈ 3.472 × 10^7 J (to three significant figures)
The net work done on the plane is equal to the change in kinetic energy, which is the difference between K2 and K1:
Net work = K2 - K1
Net work ≈ 3.472 × 10^7 J - 0 J
Net work ≈ 3.472 × 10^7 J
Therefore, the net work done by all forces on the plane as it accelerates upward is approximately 3.472 × 10^7 J.
To find the net work done by all forces on the plane, we can use the work-energy principle. The work-energy principle states that the net work done on an object is equal to the change in its kinetic energy.
The formula for work is given by W = Fd, where W is the work done, F is the force acting on the object, and d is the displacement of the object.
In this case, the force we need to consider is the net force acting on the plane. The net force can be calculated using Newton's second law of motion, F = ma, where F is the net force, m is the mass of the plane, and a is its acceleration.
Given:
Mass of the plane, m = 9.0 x 10^3 kg
Acceleration, a = 11.3 m/s^2
Displacement, d = 34.8 m
1. Calculate the net force using Newton's second law:
F = ma
= (9.0 x 10^3 kg)(11.3 m/s^2)
= 1.02 x 10^5 N
2. Calculate the work done by multiplying the force by the displacement:
W = Fd
= (1.02 x 10^5 N)(34.8 m)
= 3.55 x 10^6 J
Therefore, the net work done by all forces on the plane is 3.55 x 10^6 Joules (J).