Two simple pendulums of lengths l_1 and l_2 are oscillating with...

For a simple pendulum of length $l$ oscillating with a small angular amplitude $\theta_0$, the linear amplitude is $A = l\theta_0$.

The angular frequency of the simple pendulum is $\omega = \sqrt{\dfrac{g}{l}}$.

The maximum linear speed of the pendulum bob is given by v_\max = A\omega.

Substituting the expressions for $A$ and $\omega$:
v_\max = (l\theta_0)\sqrt{\dfrac{g}{l}} = \theta_0\sqrt{gl}

Given that the maximum linear speeds of the two pendulums are equal:
$\theta_1\sqrt{gl_1} = \theta_2\sqrt{gl_2}$

Rearranging to find the ratio of their angular amplitudes:
$\dfrac{\theta_1}{\theta_2} = \dfrac{\sqrt{l_2}}{\sqrt{l_1}} = \sqrt{\dfrac{l_2}{l_1}}$

Answer: $\sqrt{\dfrac{l_2}{l_1}}$