As shown in the figure, a block of mass m is connected to two...

When the block is displaced by a distance $x$ towards the right, the left spring is stretched by $x$ and the right spring is compressed by $x$.

Both springs exert a restoring force in the same direction (towards the left). The total restoring force on the block is:

$F = -k_1 x - k_2 x = -(k_1 + k_2)x$

This shows that the two springs act as a parallel combination with an equivalent spring constant:

$k_{eq} = k_1 + k_2$

The potential energy stored in the system at the extreme position is:

$U = \dfrac{1}{2} k_{eq} x^2 = \dfrac{1}{2} (k_1 + k_2) x^2$

When the block passes through the mean position, its potential energy is zero and its kinetic energy is maximum. By the principle of conservation of mechanical energy:

$\dfrac{1}{2} m v^2 = \dfrac{1}{2} (k_1 + k_2) x^2$

$v^2 = \dfrac{k_1 + k_2}{m} x^2$

$v = \sqrt{\dfrac{k_1 + k_2}{m}} x$