To solve the problem of a particle moving freely under the influence of gravity, we need to analyze the distances covered in two consecutive time intervals. The particle's motion can be described using the equations of uniformly accelerated motion, where the only force acting on it is gravity. Let's break this down step by step.
Understanding the Motion of the Particle
When a particle falls freely under gravity, its motion can be described by the following equations:
- The distance covered in the first time interval \( t \) is given by:
x1 = \frac{1}{2} g t^2
- The distance covered in the next time interval \( t \) (from \( t \) to \( 2t \)) is:
x2 = \frac{1}{2} g (2t)^2 - \frac{1}{2} g t^2 = \frac{1}{2} g (4t^2 - t^2) = \frac{3}{2} g t^2
Finding the Relationship Between Distances
From the above equations, we can express \( x1 \) and \( x2 \) in terms of \( t \) and \( g \) (the acceleration due to gravity). We know:
- From the first interval:
x1 = \frac{1}{2} g t^2
- From the second interval:
x2 = \frac{3}{2} g t^2
Now, we can find the difference between \( x2 \) and \( x1 \):
Calculating the Difference
The difference can be calculated as follows:
x2 - x1 = \frac{3}{2} g t^2 - \frac{1}{2} g t^2 = g t^2
Solving for Time \( t \)
Now, we want to express \( t \) in terms of \( x2 \) and \( x1 \). From the difference we derived:
g t^2 = x2 - x1
Rearranging this gives:
t^2 = \frac{x2 - x1}{g}
Taking the square root of both sides, we find:
t = \sqrt{\frac{x2 - x1}{g}}
Final Expression for Time \( t \)
Thus, the time \( t \) can be expressed as:
t = \sqrt{\frac{x2 - x1}{g}}
This equation shows how the time interval \( t \) relates to the difference in distances covered during the two intervals and the acceleration due to gravity. It’s a neat way to connect the concepts of distance and time in the context of free fall.
