I am confuse about this one question can some one please help me.
The Navy wants a new airplane launcher for their aircraft carriers and you are on the design team. The launcher is effectively a large spring that pushes the plane for the first 3.49 meters of the 24.2-meter long runway. During that same time, the plane's jet engines supply a constant thrust of 5.4 x 104 N for the entire length of the runway. The 2000 kg planes need to have a velocity of 45 m/s by the end of the runway. What should be the spring constant (in kN/m) be for the launcher?
Thank You so much!!!
To find the spring constant for the launcher, we can use the principle of work-energy theorem.
The work done by the spring is equal to the change in kinetic energy of the plane.
The work done by the spring is given by the equation:
Work_spring = (1/2) * k * x^2
Where k is the spring constant and x is the distance the spring is compressed or stretched.
The change in kinetic energy of the plane is given by:
ΔKE = (1/2) * m * (vf^2 - vi^2)
Where m is the mass of the plane, vf is the final velocity, and vi is the initial velocity.
In this case, we can assume the initial velocity (vi) is 0 since the plane starts from rest.
The work done by the spring is equal to the change in kinetic energy of the plane, so we can equate the two equations:
(1/2) * k * x^2 = (1/2) * m * vf^2
Plugging in the given values:
k * x^2 = m * vf^2
Now let's solve for x, the distance the spring is compressed or stretched.
Since the plane is pushed by the spring for the first 3.49 meters of the runway, x is equal to 3.49 meters.
Now, let's solve for the mass of the plane. It is given as 2000 kg.
Finally, the velocity of the plane at the end of the runway is given as 45 m/s.
Plugging in these values:
k * (3.49)^2 = 2000 * (45)^2
Simplifying the equation:
k * 12.1801 = 2000 * 2025
k * 12.1801 = 4,050,000
k = 4,050,000 / 12.1801
k = 332,501.23 N/m
The spring constant (k) for the launcher should be approximately 332,501.23 N/m, or 332.5 kN/m for convenience.