$2.$ A $30\mathrm{.}0$ kg child slides down a long slide in a playground. She starts from rest at a height $\backslash ($ h$\_\{1\}=25\mathrm{.}0\backslash$mathrm$\{$~m$\}\backslash ).$ When she is partway doyn the slide, at a height $\backslash ($ h$\_\{2\}=10\mathrm{.}0\backslash$mathrm$\{$~m$\}\backslash ),$ she is moving at a speed of $\backslash (5\mathrm{.}0\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$

Calculate the change in thermal energy of the system from $\backslash ($ h$\_\{1\}\backslash )$ to $\backslash ($ h$\_\{2\}\backslash ).$

a$.$ To study the long$-$term effects of weightlessness, astronauts in space must be weighed $($or at least 'massed'$).$ One way in which this is done is to seat them in a chair of known mass attached to a spring of known force constant and measure the period of the oscillations of this system. The $50\mathrm{.}0$ kg chair alone oscillates with a period of $1\mathrm{.}00$ s $,$ and the period with the astronaut sitting in the chair is $2\mathrm{.}00$ s $.$ Find the mass of the astronaut.

b$.$ An $\backslash (\backslash$mathrm$\{$m$\}=6\mathrm{.}00\backslash$mathrm$\{$~kg$\}\backslash )$ box slides into a spring of spring constant $\backslash (\backslash$mathrm$\{$k$\}=300\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash )($see figure below$),$ compressing it $0\mathrm{.}250$ m $.$ How long is the box in contact with the spring before it bounces off in the opposite direction?