How Efficient Is the Spring in Stopping a Rat on a Ski Slope?

[LoveandHate]

Homework Statement

So, I have a 5.0 kg rat that goes down a ski slope, slides along a rough patch, and then smashes into a vertical board with a spring attached. I need to find out the efficiency of the spring with this information:
m=5.0 kg
v (after sliding across rough patch)=108 m/s ; it also says that because of internal friction, the velocity is 50% than above, so it's 54 m/s (after hitting the board and spring)
$$\Delta$$x=2.42 m (the magnitude of the spring's compression)

Homework Equations

e=e_out/e_in(100%)
Possibly E_k=.5mv² and E_elastic=.5k($$\Delta$$x)²

The Attempt at a Solution

Well, I think that E_in is equal to E_k, as this is the energy required for the spring to compress, but I'm not completely sure of E^out. Is it possibly E_elastic (I know this value from a previous question)? If I use these two though, I get an answer around 400%, which is obviously wrong.

 

[PhanthomJay]

Are you saying that it hits the spring with a velocity of 104m/s, then exits the spring with a velocity of 54m/s? It might be best to state the problem as exactly worded (both parts).

 

[nlightner]

I would first clarify any uncertainties in the given information. For example, it is not clear whether the velocity given after sliding across the rough patch is the initial or final velocity before hitting the spring. Additionally, it is not specified if the velocity given after hitting the board and spring is the final velocity after the impact or the velocity at the moment of impact. These details can greatly affect the calculations and should be clarified before proceeding.

Assuming that the velocity given after sliding across the rough patch is the initial velocity before hitting the spring, and the velocity given after hitting the board and spring is the final velocity after the impact, the efficiency of the spring can be calculated using the equation e = Eout / Ein * 100%.

First, we need to calculate the initial kinetic energy (Ein) of the rat. Using the equation Ek = 0.5 * m * v^2, we get Ek = 0.5 * 5.0 kg * (108 m/s)^2 = 2916 J.

Next, we need to calculate the final kinetic energy (Eout) of the rat after hitting the board and spring. Since the velocity given is the final velocity after the impact, we can use the same equation: Ek = 0.5 * m * v^2. However, we need to use the velocity given after hitting the board and spring, which is 54 m/s. So, Eout = 0.5 * 5.0 kg * (54 m/s)^2 = 1458 J.

Now, we can calculate the efficiency of the spring using the equation e = Eout / Ein * 100%. Plugging in the values, we get e = (1458 J / 2916 J) * 100% = 50%.

This means that the spring was able to convert 50% of the rat's initial kinetic energy into elastic potential energy, and the remaining 50% was lost due to factors such as internal friction.

However, as mentioned earlier, it is important to clarify the given information to ensure accurate calculations. Additionally, it may also be helpful to consider other factors such as the mass and stiffness of the spring, and the materials and construction of the board, to get a more comprehensive understanding of the efficiency of the spring in this scenario.