Let's assume that the height from which the particles are released is h. Since both particles are released from the same height, we can consider the time interval between their releases as the time it takes for the first particle to fall h meters.
The time it takes for an object to fall from a height h can be calculated using the equation:
t = √(2h/g),
where t is the time in seconds and g is the acceleration due to gravity, which is approximately 9.8 m/s².
So, the first particle will take t seconds to fall from the height h. The second particle is released 1 second after the first particle, so it will take t - 1 seconds to fall from the same height.
After t seconds, the first particle will have fallen h meters, while the second particle will have fallen h - 10 meters because they are 10 meters apart.
Therefore, we can set up the following equation:
h - 10 = (1/2) * 9.8 * (t - 1)².
Simplifying the equation:
h - 10 = 4.9 * (t - 1)².
Expanding and rearranging the equation:
(t - 1)² = (h - 10) / 4.9.
Taking the square root of both sides:
t - 1 = √((h - 10) / 4.9).
Finally, solving for t:
t = 1 + √((h - 10) / 4.9).
Therefore, to find the time in seconds after the first particle begins to fall when the two particles are 10 meters apart, you need to substitute the value of h (the initial height) into the equation:
t = 1 + √((h - 10) / 4.9).