Answer
The speed of the gymnast at the bottom of the swing is $6.57m/s$
Work Step by Step
From the bottom of the swing to the top, the gymnast's increases by $\Delta h=1.1\times2=2.2m$
This change in height leads to an increase in PE: $\Delta PE=mg\Delta h=2.2mg (J)$
According to the principle of energy conservation, an increase in PE means a decrease in KE by the same amount: $\Delta KE=-\Delta PE=-2.2mg$
We know that $$\Delta KE=\frac{1}{2}m\Delta v^2=-2.2mg$$ $$\frac{1}{2}(v_f^2-v_0^2)=-2.2g$$ $$v_f^2-v_0^2=-4.4g=-43.12$$
When the gymnast is at the top, his speed is zero, so $v_f=0$
Therefore, $$v_0=\sqrt{v_f^2+43.12}=6.57m/s$$ which is the speed of the gymnast at the bottom of the swing.
