The $2\mathrm{.}200-$ lb carriage ; $C$ is supported as shown by two uniform $2-$lb disks, each having a radius of $3$ in $.$ Knowing that the carriage is initially at rest, determine the velocity of the carriage $0\mathrm{.}5$ s after the $0\mathrm{.}2-lb$ force $P$ has been applied. Assume that the disks roll without sliding. $($Round the final answer to four decimal places.$)$

The velocity of the carriage is $0\mathrm{.}870f\frac{t}{s}.$