Suppose if we measure the time lapse between the two Sunsets by sitting in the beach, from this we can estimate

You could simply lie down on a beach, watch the sun set once, and could quickly stand up once the last bit goes down, to see it go down again. In fact, if you carefully calculate the time difference between these 2 sunsets using a stopwatch, and take the length of your body into account, you could actually calculate (estimate) the radius of the earth.