  The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias,
Proudfoot, and Wakefield [{\it Adv. Math. 2016}]. Let $U_{m,d}$ denote the
uniform matroid of rank $d$ on a set of $m+d$ elements. Gedeon, Proudfoot, and
Young [{\it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive
an explicit formula of the Kazhdan-Lusztig polynomials of $U_{m,d}$ using
equivariant Kazhdan-Lusztig polynomials. In this paper we give two alternative
explicit formulas, which allow us to prove the real-rootedness of the
Kazhdan-Lusztig polynomials of $U_{m,d}$ for $2\leq m\leq 15$ and all $d$'s.
The case $m=1$ was previously proved by Gedeon, Proudfoot, and Young [{\it
S\'{e}m. Lothar. Combin. 2017}]. We further determine the $Z$-polynomials of
all $U_{m,d}$'s and prove the real-rootedness of the $Z$-polynomials of
$U_{m,d}$ for $2\leq m\leq 15$ and all $d$'s. Our formula also enables us to
give an alternative proof of Gedeon, Proudfoot, and Young's formula for the
Kazhdan-Lusztig polynomials of $U_{m,d}$'s without using the equivariant
Kazhdan-Lusztig polynomials.
