from what height must a heavy elastic body be dropped on a floor so that after rebounding thrice it will reach a height of 16m?take e=0.5
To solve this problem, we can apply the concept of energy conservation. When the elastic body is dropped from a certain height and rebounds, the total mechanical energy before and after the rebound should remain the same (assuming no energy is lost due to other factors like air resistance).
The total mechanical energy of the elastic body can be calculated using the formula:
Total Mechanical Energy = Potential Energy + Kinetic Energy
First, let's calculate the potential energy at the maximum height (16m):
Potential Energy (P.E) = m * g * h
where:
m = mass of the elastic body
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height (16m)
Next, let's calculate the potential energy after the first rebound:
P.E (after 1st rebound) = e * P.E (before rebound)
where:
e = coefficient of restitution (given as 0.5)
P.E (before rebound) = potential energy at the maximum height
We can use this equation to find the height after the first rebound:
P.E (after 1st rebound) = m * g * height (after 1st rebound)
Now, let's solve for the height (after 1st rebound) using the equation:
e * m * g * h = m * g * height (after 1st rebound)
Simplifying the equation:
e * h = height (after 1st rebound)
Since the height after each rebound is the same, we can repeat this process for the second and third rebounds to find the final height.
e * h = height (after 2nd rebound)
e * h = height (after 3rd rebound)
To find the initial height from which the elastic body should be dropped, we need to reverse the process. Starting from the final height (h = 16m) after the third rebound, divide it by e three times to find the initial height:
h / e / e / e = initial height
Substituting the values:
16 / 0.5 / 0.5 / 0.5 = initial height
Simplifying the equation:
16 / (0.5)^3 = initial height
16 / 0.125 = initial height
Initial Height ≈ 128m
Therefore, for a heavy elastic body to reach a height of 16m after rebounding thrice, it must be dropped from a height of approximately 128m.