HELP ALBERT PHYSICS 1 Energy of a Simple Harmonic Oscillator

Question

You are given the task of designing a dart gun powered by an ideal spring.

1. The mass, m𝑚, of the dart will be 0.030 kg.
2. The gun barrel is to be 8.0 cm long.
3. When fired horizontally, the dart is to have a velocity, v𝑣, of 4.0 m/s as it leaves the barrel of the gun.

What is the required spring constant if the spring is to be compressed the entire 8.0 cm before releasing the dart?

Please provide step by step answer so I can understand it well. Thank you!

Answer 1

Assuming (as inferred) that the spring is the same length as the barrel:

The velocity of 4 m/s as the dart leaves the barrel means we can calculate its kinetic energy:

E_K = (1/2)mv² = (1/2)(0.030)(4.0)² = 0.24 joules

When compressed, the spring would need to provide the same value of potential energy (the spring's potential energy gets converted to kinetic energy as the dart reaches the end of the barrel).

Spring Potential energy is calculated as:

E_SP = (1/2)kx² = (1/2)(k)(8/100)² = 0.0032k joules

Since E_K = E_SP then 0.24 = 0.0032k so k = 75 N/m