Question

A mass 3.00kg is attached to an ideal spring with k=200N/m, if the velocity of the body at 0.25meters is 2.3m/s. Find the amplitude and the maximum velocity

Answer 1

According to the conservation of energy, the sum of elastic potential and kinetic energies at any moment is the total mechanical energy. So, at the moment we have

$$E_p=\frac12kx^2,\space E_k=\frac12 mv^2.$$

On the other hand, when the spring is fully compressed/stretched (at the maximum amplitude), the total mechanical energy is defined by the amplitude or the maximum speed:

$$E_p+E_k=E_{p.tot}=E_{k.tot}.$$

For the amplitude:

$$\frac12kx^2+\frac12 mv^2=\frac12 kA^2,\\\space\\ A=\sqrt{x^2+\frac{mv^2}{k}}=0.377\text{ m}.$$

For the maximum velocity:

$$\frac12kx^2+\frac12 mv^2=\frac12 mV^2,\\\space\\ V=\sqrt{\frac{kx^2}{m}+v^2}=3.08\text{ m/s}.$$