Please reason step by step, and put your final answer within \boxed{}.
An object moves in a straight line in a certain medium. At time $t$, its velocity is $v$, and its displacement is $s$. The resistance $F$ it experiences in this medium is directly proportional to the square of its velocity $v$, with a proportionality constant $k$. Given that $s= \frac{1}{2}t^2$, the work done by the object to overcome the resistance from displacement $s=0$ to displacement $s=a$ is \_\_\_\_\_\_ . (Note: The work $W$ done by a variable force $F$ is $W=\int_{s_1}^{s_2} F(s) \, ds$, and the result should be expressed in terms of $k$ and $a$.)
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