Answer
The force must act for 93 days.
Work Step by Step
We can use a force equation to find the acceleration.
$ma = F$
$a = \frac{F}{m}$
$a = \frac{2.5 ~N}{1.0 \times 10^{10} ~kg}$
$a = 2.5 \times 10^{-10} ~m/s^2$
We can use kinematics to find the required time.
$a = \frac{\Delta v}{t}$
$t = \frac{\Delta v}{a}$
$t = \frac{2.0 \times 10^{-3} ~m/s}{2.5 \times 10^{-10} ~m/s^2}$
$t = 8.0 \times 10^6~s$
We can convert this time from seconds to days.
$t = (8.0 \times 10^6~s)(\frac{1 ~h}{3600 ~s})(\frac{1 ~day}{24 ~h})$
$t = 93 ~days$
The force must act for 93 days.
