Answer
$1.6\,m/s^{2}$
Work Step by Step
The maximum tension that the rope can withstand is 425 N.
By setting the tension or force equal to 425 N, we can find the magnitude of the acceleration of circus performer that just avoid breaking the rope.
$F=m(g-a)=425\,N$
$\implies 52\,kg(9.8\,m/s^{2}-a)=425\,N$
$\implies 510\,N-52\,kg\times a=425\,N$
Or $a\times52\,kg=510\,N-425\,N=85\,N$
Or $a=\frac{85\,N}{52\,kg}=1.6\,m/s^{2}$
