Help Spring Avarage acceleration and Spring Constant in a Spacecraft

[yogui]

Help Spring Avarage acceleration and Spring Constant in a Spacecraft !

Homework Statement

In NASA’s current concept, the lander has 4 legs, each with a plate on the end that will set firmly on the surface. Each leg has a spring to act as a shock absorber and each shock will compress 30 cm when the leg plates touch the surface at a velocity of 0.5 m/s. Assume the engines’ thrust goes instantly to zero at the time of contact with the surface and the lunar surface does not compress. What is the average acceleration of the vehicle after it touches the surface? What is the spring constant for each spring?


V = .5 m/s
M= 20,000 Kg

Homework Equations

F=Kd Hookes law
d= 1/2(Vf+Vi)T
F=ma

The Attempt at a Solution

What i would do is: F=Kd
F=ma
ma= K (.30 m)
20,000 ( ∆V/T)= K (.30)
∆V/T = (Vf-Vi )/t = ( 0 m/s - .5 m/s )/T

D= 1/2 (Vf+Vi) T
T= 2D/ (Vf+Vi)
T = 2 ( .30 m) / (.5 m/s)
T=1.2 s

replacing..
∆V/T = ( 0 m/s - .5 m/s )/ 1.2 s
= .416 m/s^2

20,000 ( ∆V/T)= K (.30)
20,000 ( .416)= K (.30)
8,333 N / .30 m = k

K= 27,777.77 N/m

it gave me a big number, I am not sure if my process is right because i have not taken this at my physics class yet, but based on reading that's what i get.

As for the average aceleration I am not sure.
i would say
a= ∆V/T = ( 0 m/s - .5 m/s )/ 1.2 s
a = .416 m/s^2
Any help is greatly appreciated, Thankz!

 

[anathema]

I have reviewed your solution and it seems to be mostly correct. However, there are a few things that I would like to clarify and suggest for improvement.

Firstly, your use of Hooke's law (F=Kd) is correct, but you have not taken into account the weight of the spacecraft in your calculation. The total force acting on the spring will be the sum of the weight of the spacecraft (mg) and the force exerted by the shock (F). So your equation should be:

mg + F = Kd

Also, in your calculation for the spring constant, you have used the wrong value for ∆V/T. The correct value should be (Vf-Vi)/T, which in this case is (0.5 m/s)/1.2 s = 0.4167 m/s^2. This will give you a slightly different value for the spring constant.

For the average acceleration, you have calculated it correctly. The average acceleration of the spacecraft after it touches the surface will be 0.4167 m/s^2.

I would also like to suggest a different approach to solving this problem. Instead of using the formulas for Hooke's law and motion, you can use the principle of conservation of energy to solve for the spring constant. The total energy of the system (kinetic + potential) will remain constant, so you can equate the initial and final energies to solve for the spring constant.

Initial energy = 1/2mv^2
Final energy = 1/2kx^2

Since the velocity becomes zero after the shock is compressed, the final energy will be zero. So you can equate the initial energy to zero and solve for the spring constant. This will give you a more accurate value for the spring constant.

I hope this helps. Let me know if you have any further questions. Good luck with your physics studies!

 

[nlightner]

Hello, thank you for your question. It seems like you have the right idea in terms of using Hooke's Law and Newton's Second Law to solve for the spring constant and average acceleration. However, there are a few things to consider in your calculations.

First, it is important to note that the given information does not specify the time it takes for the leg plates to compress and reach a velocity of 0.5 m/s. Therefore, it is not accurate to say that the time of contact with the surface is 1.2 seconds. Instead, we can assume that the time taken for the legs to compress and reach 0.5 m/s is very short and can be considered instantaneous.

Second, in your calculation for the spring constant, you have used the value of 20,000 kg for the mass of the spacecraft. However, this value is not relevant to the calculation as it is not involved in the forces acting on the legs. The only relevant forces are the spring force and the weight of the spacecraft. Therefore, the mass should not be included in the calculation for the spring constant.

With these considerations, the correct calculation for the spring constant should be:

F = Kd
mg = K(.30 m)
K = mg/.30 m = (20,000 kg)(9.8 m/s^2)/.30 m = 653,333.33 N/m

For the average acceleration of the vehicle, you have correctly calculated it to be 0.416 m/s^2. This is the average acceleration during the time it takes for the legs to compress and reach a velocity of 0.5 m/s. After that, the acceleration will decrease as the legs continue to compress and the spacecraft comes to a stop on the surface.

I hope this helps clarify the calculations for you. Let me know if you have any further questions. Good luck!