In this article, we study the family of elliptic curves given by Ep:y2=x3-5px for an odd prime p≠5 and try to find the conditions under which Ep has rank one. Specifically, we have shown that if p≡3,27(mod40) such that 2(p+5) is a perfect square or if p≡11,19(mod40) such that (5+4p) is a perfect square, then the Mordell-Weil rank of Ep is exactly one. Finally, assuming the parity conjecture, we have proved that if p≡3,11,19,27(mod40), then the rank of Ep is always one.