The ratio of energy required to accelerate a car from rest to 20 ms^–1 to the energy needed to accelerate from 20ms^–1 to 40ms^–1 is

The energy required to increase its speed from 0 to 20m/s is given by 

${\mathrm{E}}_1=\frac{1}{2}\mathrm{m}({20}^2-0^2)=\frac{1}{2}\mathrm{m}\left(400\right)$

The energy required to accelerate from 20m/s to 40m/s

$E_2=\frac{1}{2}m({40}^2-{20}^2)=\frac{1}{2}m\left(1200\right)$

Thus ratio,

$$\begin{gathered} ratio=\frac{E_1}{E_2}=\frac{{\displaystyle \frac{1}{2}}m\left(400\right)}{{\displaystyle \frac{1}{2}}m\left(1200\right)} \\ {E}_1:E_2=1:3 \end{gathered}$$

Hence the correct answer is 1:3.