To tackle this problem, let's break it down step by step, using the concepts of free fall and kinematics. When a stone falls freely from rest, its motion can be described using the equations of uniformly accelerated motion, where the acceleration is due to gravity, typically taken as approximately 9.81 m/s². We want to find the total time the stone remains in the air, given a specific relationship between distances covered at different times.
The Relationship Between Distances
According to the problem, the distance covered by the stone in the last second of its fall is equal to the distance it covered in the first three seconds. Let’s denote:
- t = total time of fall (in seconds)
- s(t) = distance covered in t seconds
- s(3) = distance covered in the first 3 seconds
- s(t) - s(t-1) = distance covered in the last second
Calculating Distances
The distance covered by an object in free fall from rest can be calculated using the formula:
s(t) = (1/2) * g * t²
Where g is the acceleration due to gravity (approximately 9.81 m/s²).
For the first three seconds:
s(3) = (1/2) * g * (3)² = (1/2) * 9.81 * 9 = 44.145 m
Now, for the total distance covered in t seconds:
s(t) = (1/2) * g * t²
The distance covered in the last second can be expressed as:
s(t) - s(t-1) = (1/2) * g * t² - (1/2) * g * (t-1)²
This simplifies to:
g * (t - 0.5)
Setting Up the Equation
According to the problem, we set the distance covered in the last second equal to the distance covered in the first three seconds:
g * (t - 0.5) = 44.145
Substituting the value of g:
9.81 * (t - 0.5) = 44.145
Solving for Time
Now, we can solve for t:
t - 0.5 = 44.145 / 9.81
t - 0.5 ≈ 4.5
t ≈ 5
Conclusion
This tells us that the stone remains in the air for approximately 5 seconds. Therefore, the correct answer is (B) 5 s.
Understanding the motion of falling objects and applying the equations of motion helps us derive such conclusions effectively. It's fascinating how these principles can be applied to solve real-world problems in physics!
