Time, velocity, deceleration and the constant K

[Northbysouth]

Homework Statement

A certain lake is proposed as a landing area for large jet aircraft. The touchdown speed of 110 mi/hr upon contact with the water is to be reduced to 24 mi/hr in a distance of 1370 ft. If the deceleration is proportional to the square of the velocity of the aircraft through the water, a = -Kv2, find the value of the design parameter K, which would be a measure of the size and shape of the landing gear vanes that plow through the water. Also find the time t elapsed during the specified interval.

Homework Equations

x = x₀ + v_0xt + 1/2at²

v = v_0x + at

The Attempt at a Solution

v = v_0x + at

24 mi/hr = 110 mi/hr - (kv²)t
24 mi/hr = 110 mi/hr -576kt
576kt = 86 mi/hr

t = 86/576k

Taking this I plugged it into

x = x₀ + v_0xt + 1/2at²

1370ft = 0 + (110mi/hr)t + 1/2(-kv²)t²

Plugged in v = 24 mi/hr

1370ft = (110 mi/hr)5 - 288 kt²

1370 ft = (110 mi/hr)(86/576k) - 288(86/576)

k = 0.01162

t = 12/845

I realize that I entered my value for k into the system incorrectly, but I used that value to calculate t, which was incorrect. With that said, I'm not sure what to do. Suggestions are welcome and appreciated. THanks

 

[tms]

Homework Statement

A certain lake is proposed as a landing area for large jet aircraft. The touchdown speed of 110 mi/hr upon contact with the water is to be reduced to 24 mi/hr in a distance of 1370 ft. If the deceleration is proportional to the square of the velocity of the aircraft through the water, a = -Kv2, find the value of the design parameter K, which would be a measure of the size and shape of the landing gear vanes that plow through the water. Also find the time t elapsed during the specified interval.

Homework Equations

x = x₀ + v_0xt + 1/2at²

v = v_0x + at

The Attempt at a Solution

v = v_0x + at

24 mi/hr = 110 mi/hr - (kv²)t
24 mi/hr = 110 mi/hr -576kt
576kt = 86 mi/hr

It is best to solve all problems symbolically before plugging in any numbers. Each time one plugs in a number, one loses information that can help solve the problem or check one's answer for sanity.

However, your difficulty here is that the equations you are using only apply when the acceleration is constant, whereas in this problem the acceleration depends on velocity. You have to start from first principles, that is, the definitions of velocity and acceleration.

 

[Northbysouth]

When you say first principles do you mean:

v = dx/dt

a = dv/dt = dv/dx*dx/dt = v dv/dx

Will I need differential equations to solve this problem?

 

[tms]

When you say first principles do you mean:

v = dx/dt

Yes. This one is correct.

a = dv/dt = dv/dx*dx/dt = v dv/dx

This one starts out correctly, but then goes astray:
$$a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$$

Will I need differential equations to solve this problem?

No; simple integration is enough.

You are given
$$a = \frac{dv}{dt} = -Kv^2.$$
Start from there.

 

[Nickie]

!

I would like to point out that the design parameter K is not solely a measure of the size and shape of the landing gear vanes, but also a measure of the drag and resistance of the aircraft through water. The value of K will depend not only on the size and shape of the vanes, but also on the specific design of the aircraft and its materials.

In terms of the solution, it seems that you have made some errors in your calculations. I would suggest revisiting your equations and making sure to use consistent units (miles vs. feet) throughout. Additionally, you may want to double check your substitution of values and use of the quadratic formula. It may also be helpful to plot the given information on a graph and use the slope to find the value of K.

Overall, this is an interesting problem that highlights the importance of considering various factors when designing a landing area for aircraft. As a scientist, it is important to carefully consider all aspects and variables in a problem and make accurate calculations based on sound equations and principles.