When the gymnast is at 2.98 meters above the trampoline, she has potential gravitational energy in relation to the trampoline:
[tex]\begin{gathered} PE_g=m\cdot g\cdot h \\ PE_g=52\cdot9.81\cdot2.98 \\ PE_g=1520.16\text{ J} \end{gathered}[/tex]When she lands on the trampoline, all this energy will be converted into potential elastic energy:
[tex]\begin{gathered} PE_e=\frac{kx^2}{2} \\ 1520.16=\frac{k\cdot0.7^2}{2} \\ 0.49k=3040.32 \\ k=6204.73\text{ N/m} \end{gathered}[/tex]Now, to find the stretch in the trampoline when the gymnast is at rest, let's use the gymnast weight force in the formula for the force in a string:
[tex]\begin{gathered} F=k\cdot x \\ m\cdot g=k\cdot x \\ 52\cdot9.81=6204.73\cdot x \\ 510.12=6204.73 \\ x=\frac{510.12}{6204.73} \\ x=0.0822\text{ m} \end{gathered}[/tex]Therefore the trampoline stretches 8.22 cm.
