A stone falls freely from rest and the total distance covered by it in...
Understanding the Scenario
To solve this problem, let's first understand the scenario. We have a stone that falls freely from rest. This means that the stone is only under the influence of gravity, and there are no other forces acting on it. We need to determine how long the stone remains in the air.
Using Equations of Motion
We can use the equations of motion to solve this problem. The equation that relates distance, time, and acceleration for an object in free fall is:
s = ut + (1/2)at^2Where:
- s is the distance covered
- u is the initial velocity (which is zero as the stone starts from rest)
- t is the time taken
- a is the acceleration due to gravity (which is approximately 9.8 m/s^2)
Calculating the Distance
Let's break down the problem into two parts: the first three seconds and the last second.
First Three Seconds:Using the equation of motion, we can calculate the distance covered by the stone in the first three seconds:
s = 0 + (1/2)(9.8)(3)^2s = 0 + (1/2)(9.8)(9)s = 0 + 44.1s = 44.1 metersLast Second:Now, we know that the distance covered in the last second is equal to the distance covered in the first three seconds. Let's denote this distance as 'd'.
d = 44.1 metersCalculating the Time
We can rearrange the equation of motion to solve for time:
s = ut + (1/2)at^2t^2 + 2(s/a)t - 2(0) = 0Let's substitute the values:
t^2 + (2d/a)t = 0Now, we can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2aWhere:
- a = 1
- b = 2d/a = 2d/9.8
- c = 0
t = (-2d/9.8 ± √((2d/9.8)^2 - 4(1)(0))) / 2(1)t = (-2d/9.8 ± √((4d^2/9.8^2))) / 2t = (-2d/9.8 ± (2d/9.8)) / 2t = -d/9.8 + d/9.8t = 0Conclusion
From our calculation, we find that the time taken for the stone to cover the distance 'd' in the last second is zero. This means that the stone remains in the air for zero seconds in the last second of its motion. Therefore