Calculating Maximum Height and Final Velocity: 2 Energy Problems on a Slope

[rain_ex]

Homework Statement

A 50g ice cube has no friction. It compresses a string with the constant 25 N*m a total of 10 cm. Find the max height reached by the ice cube before it starts sliding back down the slope, after it is released from the spring.

Known:
m = 50g (.05kg)
k = 25 Nm
x = 10cm (.1m)

Homework Equations

E₀ = E <- general
Us₀ = Ug <- expanded
1/2kx² = mgh <- final equation

The Attempt at a Solution

I ended up with 0.255m. Is this correct? I solved for the variable "h".

Homework Statement

A cannon angled 30 degrees above the horizontal launches a cannanball at 80 m/s. if the cannon is 10 m above the ground with what speed does the cannon ball strike the ground?

Homework Equations

E₀ = E <- general
Ug₀ + K₀ = K <- expanded
mgh+1/2mv₀² = 1/2mv² <-final equation

The Attempt at a Solution

I ended up with 81.216 m/s. I solved for the variable "v".

Please check my solutions for me! Thank you!

 

[ehild]

Your solution is correct, but the unit of the spring constant k is N/m instead of Nm.

ehild

 

[rain_ex]

Thank you.
I'm assuming that the 2nd question's solution was correct too?

 

[ehild]

Yes, both solutions are correct.

ehild

 

[rwisz]

Your solutions appear to be correct. In the first problem, you correctly used the conservation of energy equation to solve for the maximum height reached by the ice cube. In the second problem, you used the conservation of energy equation again, taking into account the initial potential and kinetic energies to solve for the final velocity of the cannonball before it strikes the ground. Overall, your approach and solutions demonstrate a good understanding of energy conservation principles. Keep up the good work!