The figure displays two wave pulses on a string at t = 0 ,...

The initial position of the first wave pulse (positive displacement) is from $x = 2 \text{ mm}$ to $x = 6 \text{ mm}$. It is travelling in the positive x-direction.

The initial position of the second wave pulse (negative displacement) is from $x = 10 \text{ mm}$ to $x = 14 \text{ mm}$. It is travelling in the negative x-direction.

The wave speed is given as $v = 50 \text{ cm s}^{-1} = 500 \text{ mm s}^{-1}$.

At any time $t$, the position of the first pulse is $[2 + vt, 6 + vt]$ and the position of the second pulse is $[10 - vt, 14 - vt]$.

For the string to have zero displacement everywhere, the positive pulse must completely overlap with the negative pulse. This requires their positions to be identical:

$2 + vt = 10 - vt$

$2vt = 8 \implies vt = 4 \text{ mm}$

Substituting the value of $v$:

$500 \times t = 4$

$t = \dfrac{4}{500} \text{ s} = 0.008 \text{ s}$

Converting to milliseconds:

$t = 8 \text{ ms}$

At $t = 8 \text{ ms}$, both pulses exactly overlap between $x = 6 \text{ mm}$ and $x = 10 \text{ mm}$, cancelling each other out completely.