Answer
A tension of 108 N will give a speed of 180 m/s
Work Step by Step
We can write an expression for the initial speed which is 150 m/s:
$v_1 = \sqrt{\frac{T_1}{\mu}}$
We can write an expression for the final speed which is 180 m/s:
$v_2 = \sqrt{\frac{T_2}{\mu}}$
We can divide the second equation by the first equation.
$\frac{v_2}{v_1} = \frac{\sqrt{\frac{T_2}{\mu}}}{\sqrt{\frac{T_1}{\mu}}}$
$\frac{\sqrt{T_2}}{\sqrt{T_1}} = \frac{v_2}{v_1}$
$\sqrt{T_2} = \sqrt{T_1}~\frac{v_2}{v_1}$
$T_2 = T_1~(\frac{v_2}{v_1})^2$
$T_2 = (75.0~N)~(\frac{180~m/s}{150~m/s})^2$
$T_2 = 108~N$
A tension of 108 N will give a speed of 180 m/s.
