#### Answer

(a) We will have to push for 2.0 seconds to reach the speed $v$.
(b) We will have to push for 1.41 seconds for the puck with mass $2m$ to travel a distance $d$.

#### Work Step by Step

(a) We know that $F = ma$
Let $a_1$ be the acceleration of the puck with mass $m$. We can find an expression for $a_1$.
$F = m~a_1$
$a_1 = \frac{F}{m}$
Let $a_2$ be the acceleration of the puck with mass $2m$. We can find an expression for $a_2$.
$F = (2m)(a_2)$
$a_2 = \frac{F}{2m} = \frac{a_1}{2}$
We can find an expression for $v$ in terms of $a_1$
$v = a_1~t = (1.0~s)~a_1$
We can find the time for the puck with mass $2m$ to reach the speed $v$.
$v = a_2~t = (1.0~s)~a_1$
$(\frac{a_1}{2})~t = (1.0~s)~a_1$
$t = 2\times (1.0~s)$
$t = 2.0~s$
We will have to push for 2.0 seconds to reach the speed $v$.
(b) We can find an expression for $d$ in terms of $a_1$.
$d =\frac{1}{2}~a_1t^2$
$d =\frac{1}{2}~a_1~(1.0~s)^2$
We can find the time for the puck with mass $2m$ to travel a distance $d$.
$d = \frac{1}{2}a_2t^2 = \frac{1}{2}~a_1~(1.0~s)^2$
$\frac{a_1}{2}~t^2 = a_1~(1.0~s)^2$
$t = (1.0~s)~\sqrt{2}$
$t = 1.41~s$
We will have to push for 1.41 seconds for the puck with mass $2m$ to travel a distance $d$.