A body A starts from rest with an acceleration a1. After 2 seconds, another body B starts from rest with an acceleration a2 . If they travel equal distances in the 5th second, after the start of A, then the ratio a1 : a2 is equal to

The correct option is 5:9.

Let the two bodies be A and B. Both start from rest, which means their initial velocities are zero:
$u_A=0$
and
$u_B=0$

The distance traveled by a body in its n^th second of motion is given by the formula:
$S_n=u+\frac{a}{2}(2n-1)$

For body A, we calculate the distance traveled in the 5^th second after its start (which corresponds to n = 5):
$S_{5,A}=0+\frac{a_1}{2}\left(2\right(5)-1)=\frac{9a_1}{2}$

Body B starts 2 seconds after body A. Therefore, the 5^th second after the start of A corresponds to the 3^rd second of motion for body B (since 5 - 2 = 3). For body B, we set n = 3:
$S_{3,B}=0+\frac{a_2}{2}\left(2\right(3)-1)=\frac{5a_2}{2}$

According to the problem, the distances traveled by both bodies during this second are equal:
$S_{5,A}=S_{3,B}$
Substituting the values derived above:
$\frac{9a_1}{2}=\frac{5a_2}{2}$

Simplifying the equation gives:
$9a_1=5a_2$
Rearranging the equation to find the ratio of their accelerations:
$\frac{a_1}{a_2}=\frac{5}{9}$
Therefore, the ratio of their accelerations a1 : a2 is 5:9.