$7.$ Consider a rocket in empty space.

$($a$)$ If a rocket motor's exhaust speed is increased by a factor of $2,$ does the rocket's

maximum speed increase by a factor of $2,$ more than $2,$ or less than $2?$ Explain.

$($b$)$ If a rocket's fuel$-$to$-$empty$-$rocket mass ratio is increased by a factor of $2,$ does the

rocket's maximum speed increase by a factor of $2,$ more than $2,$ or less than $2?$

Explain.