Need help with problems involving springs

[Winged]

I am in a Physics I class and need help with my homework. There are a couple problems involving springs that I don't know how to do. I think if I can figure out how to do one of them, I would be able to do the other, so I'll just post one question for now.

Homework Statement

A ball is compressed against a vertical spring by 1 m. The ball has a mass of 10 kg. The spring has a constant of 10,000 N/m. The ball is 4 meters above the ground. How high will the ball travel and how fast will it hit the ground? Assume you have moved the spring in time.

Homework Equations

I have two equations that our professor gave us that are used for springs. They are:
PE_s= $$\frac{1}{2}$$kx²

T=2$$\Pi$$$$\sqrt{\frac{m}{k}}$$

I'm pretty sure there are one or two other equations I need to use for this problem, but I don't know what they are. I think this is where my problem is.

The Attempt at a Solution

So far, the only thing I've done is what I knew how to do.
PE_s= $$\frac{1}{2}$$kx²
PE_s=5000

T=2$$\Pi$$$$\sqrt{\frac{m}{k}}$$
T=0.1987 (I don't think this is really needed for solving the equation, but I really have no idea, and I had to start somewhere.)

Any help would be greatly appreciated! Thanks!

 

[PhanthomJay]

Have you studied conservation of energy?

 

[Winged]

Yes, we have. Looking in my notes, I have one equation involving the conservation of energy, and it is:

W_c=F_cd=-$$\Delta$$PE=-(PE_f-PE₀

 

[PhanthomJay]

Yes, we have. Looking in my notes, I have one equation involving the conservation of energy, and it is:

W_c=F_cd=-$$\Delta$$PE=-(PE_f-PE₀

That equation tells you that the work done by conservative forces is the negative of the potential energy change. For part 1, try $$\Delta K + \Delta U = 0$$ .
What is delta K when the block travels from rest to its max height?

 

[rwisz]

Dear student,

Thank you for reaching out for help with your physics homework. It seems like you have a good understanding of the equations for springs, but just need a little guidance on how to apply them to this specific problem.

To start, let's break down the problem into smaller parts. First, we need to find the initial potential energy (PEi) of the spring when it is compressed by 1 m. This can be calculated using the equation you provided: PEs = 1/2*k*x^2. Plugging in the given values, we get PEs = 1/2 * 10,000 * (1)^2 = 5,000 J.

Next, we need to find the initial kinetic energy (KEi) of the ball. This can be calculated using the equation KEi = 1/2 * m * v^2, where m is the mass of the ball and v is its initial velocity. In this case, the ball is initially at rest, so its initial velocity is 0. Therefore, KEi = 1/2 * 10 * (0)^2 = 0 J.

Now, we can use the principle of conservation of energy to find the final energy of the system. This means that the initial energy (PEi + KEi) must equal the final energy (PEf + KEf). Since we know the initial potential and kinetic energies, we can solve for the final potential energy (PEf) and kinetic energy (KEf).

PEf = PEi = 5,000 J

To find KEf, we can use the equation KEf = 1/2 * m * v^2, where m is still 10 kg and v is the final velocity of the ball before it hits the ground. To find this velocity, we can use the equation T = 2π√(m/k), where T is the period of oscillation (the time it takes for the ball to complete one cycle of going up and down on the spring). We can solve for T by setting the initial potential energy to the final potential energy and solving for T.

5,000 J = 1/2 * 10,000 N/m * (T)^2
T = √(1/10) s = 0.3162 s

Now, we can plug this value for T into the equation for KEf to solve for the final kinetic energy