Over the $0\mathrm{.}200$ second signal, the frequency increased from $35\mathrm{.}0$ Hz to $150$ Hz$,$ at which point the merger took place, and there was a short, characteristic "ringdown" to zero signal. Using general relativity, it can be determined that the effective relative speed of the inwardly spiraling black holes is given by

v $=($GM$$f$)1/3,$

where M is the total mass of the system $($before merger$)$ and f is the gravitational wave frequency. When the frequency of the signal was $50\mathrm{.}0$ Hz$,$ what was the relative speed of the black holes as a fraction of the speed of light?

vc

$=$