A force of 1N acts on a 1kg mass at rest for 1s. In another case 1N force acts on 1kg mass at rest and moves it through 1m. The ratio of kinetic energies in the two cases is

In the first case Let v₁ be the final velocity of the mass

F.t = mv_f - mv_i
⇒ 1 x 1= 1 x v₁ - 1 x 0
⇒ v₁ = 1m/s.
In second case, Let v₂ be the final velocity of the mass
Now, 

In the first case Let v1 be the final velocity of the massF.t = mvf - mvi⇒ 1 x 1= 1 x v1 - 1 x 0⇒ v1 = 1m/s.In second case, Let v2 be the final velocity of the massNow,$W_{net}=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

$\Rightarrow 1\times 1=\frac{1}{2}\times v_2^2-0$

$\Rightarrow v_2=\sqrt{2} \mathrm{m}/\mathrm{s}$

$\frac{(KE{)}_1}{(KE{)}_2}=\frac{1/2\times 1\times (1{)}^2}{1/2\times 1\times (\sqrt{2}{)}^2}=\frac{1}{2}$