(a)
The work done by the spring force can be calculated with the formula below:
[tex]W=\frac{1}{2}kx^2[/tex]
Using k = 320 N/m and x = 0.075 m, we have:
[tex]\begin{gathered} W=\frac{1}{2}\cdot320\cdot0.075^2\\ \\ W=0.9\text{ J} \end{gathered}[/tex]
(b)
The increase in thermal energy is given by the work done by the friction force.
To calculate this work, first let's find the friction force:
[tex]\begin{gathered} F_{friction}=F_{normal}\cdot\mu\\ \\ F_{friction}=m\cdot g\cdot\mu\\ \\ F_{friction}=2.5\cdot9.8\cdot0.25\\ \\ F_{friction}=6.125\text{ N} \end{gathered}[/tex]
Now, calculating the work, we have:
[tex]\begin{gathered} W=F\cdot d\\ \\ W=6.125\cdot0.075\\ \\ W=0.46\text{ J} \end{gathered}[/tex]
(c)
The block speed can be found by converting the potential energy from the spring (same value of the calculated work in item a) into kinetic energy for the block:
[tex]\begin{gathered} PE=KE\\ \\ 0.9=\frac{mv^2}{2}\\ \\ mv^2=1.8\\ \\ 2.5v^2=1.8\\ \\ v^2=\frac{1.8}{2.5}\\ \\ v^2=0.72\\ \\ v=0.8485\text{ m/s} \end{gathered}[/tex]
