Answer:
The maximum length during the motion is ![L_{max} = 1.45m](https://tex.z-dn.net/?f=L_%7Bmax%7D%20%3D%201.45m)
Explanation:
From the question we are told that
The mass is ![m =0.5 kg](https://tex.z-dn.net/?f=m%20%3D0.5%20kg)
The vertical spring length is ![L = 1.10m](https://tex.z-dn.net/?f=L%20%3D%201.10m)
The unstretched length is ![L_{un} = 1.30m](https://tex.z-dn.net/?f=L_%7Bun%7D%20%3D%201.30m)
The initial speed is ![v_i = 1.3m/s](https://tex.z-dn.net/?f=v_i%20%3D%201.3m%2Fs)
The new length of the spring ![L_{new} = 1.30 m](https://tex.z-dn.net/?f=L_%7Bnew%7D%20%3D%20%201.30%20m)
The spring constant k is mathematically represented as
![k = -\frac{F}{y}](https://tex.z-dn.net/?f=k%20%3D%20-%5Cfrac%7BF%7D%7By%7D)
Where F is the force applied ![= m * g = 0.5 * 9.8=4.9N](https://tex.z-dn.net/?f=%3D%20m%20%2A%20g%20%3D%200.5%20%2A%209.8%3D4.9N)
y is the difference in weight which is ![=1.10-0.50=0.6m](https://tex.z-dn.net/?f=%3D1.10-0.50%3D0.6m)
The negative sign is because the displacement of the spring (i.e its extension occurs against the force F)
Now substituting values accordingly
![k = \frac{4.9}{0.6}](https://tex.z-dn.net/?f=k%20%3D%20%20%5Cfrac%7B4.9%7D%7B0.6%7D)
![= 8.17 N/m](https://tex.z-dn.net/?f=%3D%208.17%20N%2Fm)
The elastic potential energy is given as ![E_{PE} = \frac{1}{2} k D^2](https://tex.z-dn.net/?f=E_%7BPE%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20k%20D%5E2)
where D is this the is the displacement
Since Energy is conserved the total elastic potential energy would be
![E_T = initial \ elastic\ potential \ energy + kinetic \ energy](https://tex.z-dn.net/?f=E_T%20%3D%20initial%20%20%5C%20elastic%5C%20potential%20%5C%20energy%20%2B%20kinetic%20%5C%20energy)
Substituting value accordingly
![\frac{1}{2} *8.17 *D_{max}^2 =\frac{1}{2} * 8.17*(1.30 - 0.50)^2 + \frac{1}{2} * 0.5 *1.30^2](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%2A8.17%20%2AD_%7Bmax%7D%5E2%20%3D%5Cfrac%7B1%7D%7B2%7D%20%2A%208.17%2A%281.30%20-%200.50%29%5E2%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%2A%200.5%20%2A1.30%5E2)
![4.085 * D_{max}^2 = 3.69](https://tex.z-dn.net/?f=4.085%20%2A%20D_%7Bmax%7D%5E2%20%3D%203.69)
![D^2_{max} = 0.9033](https://tex.z-dn.net/?f=D%5E2_%7Bmax%7D%20%3D%200.9033)
![D_{max} = 0.950m](https://tex.z-dn.net/?f=D_%7Bmax%7D%20%3D%200.950m)
So to obtain total length we would add the unstretched length
So we have
![L_{max} = 0.950 + 0.5 = 1.45m](https://tex.z-dn.net/?f=L_%7Bmax%7D%20%3D%200.950%20%2B%200.5%20%3D%201.45m)