The Navy wants a new airplane launcher for their aircraft carriers that is basically a large spring and your job is to determine the necessary spring constant. The launcher pushes the plane for a short distance along a much longer runway. During that same time, the plane's jet engines supply a constant thrust force for the entire length of the runway. The planes need to have a minimum velocity by the time they reach the end of the runway in order to take off successfully.
(a) What is the algebraic expression for the spring constant of the launcher in terms of the mass of the plane (M), the minimum take-off speed of the plane (V), the thrust force by the engines (F), the length of the runway (L), the distance that the spring pushes the plane (D), and the gravitational field strength (g)? [Note: Don't enter an equation like "x=blah". Just enter the "blah" part. All letters are capital except for "g".]
To determine the necessary spring constant for the airplane launcher, we need to consider the forces acting on the plane.
First, we can consider the work done by the spring launcher. The work done by a spring is given by the equation:
Work = (1/2)kx^2
where k is the spring constant and x is the distance the spring pushes the plane (D in this case).
The work done by the spring must be equal to the work done by the engines in order to achieve the minimum take-off speed. The work done by the engines is given by the equation:
Work = Force x Distance
Since the thrust force by the engines is constant and the distance along the runway is L, the work done by the engines can be expressed as:
Work = Force x L
Setting these two equations equal to each other, we have:
(1/2)kx^2 = Force x L
Now, we can consider the force of the engines. The engines provide a constant thrust force, which is equal to the mass of the plane (M) multiplied by the acceleration due to gravity (g):
Force = M x g
Substituting this into our equation, we get:
(1/2)kx^2 = (M x g) x L
Finally, we can rearrange the expression to solve for the spring constant (k):
k = (2 x M x g x L) / x^2
So, the algebraic expression for the spring constant of the launcher in terms of the given variables is:
k = (2 x M x g x L) / x^2