An object of mass 2 kg is dropped from a certain
height. On rebounding from the ground, it rises
vertically till
2
th
5
of its initial height. The ratio of
magnitude of momentum of the object just before
and after striking the ground is
The answer is √5:√2
Yes
My answer is root5:root2
To find the ratio of the magnitude of momentum just before and after striking the ground, we need to first calculate the momentum before and after the rebound.
Let's break down the problem step by step:
Step 1: Find the initial momentum before striking the ground.
The magnitude of momentum can be calculated using the equation:
Momentum = Mass x Velocity
Given:
Mass of the object (m) = 2 kg
Since the object is just dropped, it does not have any initial velocity. Therefore, the initial momentum is zero.
Initial momentum (before striking the ground) = 0 kg·m/s
Step 2: Find the final momentum just before striking the ground.
Since the object rebounds after striking the ground, we need to find the velocity just before striking the ground.
Given:
The object rises vertically till 2/5 of its initial height.
Let's assume the initial height is 'h'. Then the height just before striking the ground is (2/5)h.
Now, we can use the law of conservation of mechanical energy to find the velocity just before striking the ground:
Initial potential energy = Final kinetic energy
Potential energy before striking the ground = mgh
Kinetic energy just before striking the ground = (1/2)mv^2, where v is the velocity just before striking the ground.
Since the object was dropped, it had zero initial velocity. Therefore, the potential energy before striking the ground is equal to the initial potential energy:
mgh = (1/2)mv^2
Canceling out the mass 'm', we get:
gh = (1/2)v^2
Now, substituting the value (2/5)h for the height just before striking the ground, we get:
g(2/5)h = (1/2)v^2
Simplifying:
v^2 = (2gh)(5/2) / 1
v^2 = 5gh
Taking the square root of both sides, we get:
v = √(5gh)
The final momentum just before striking the ground is:
Momentum = Mass x Velocity
Momentum = 2 kg x √(5gh)
Step 3: Find the momentum after the rebound.
Since the object rebounds, the velocity after the rebound is equal in magnitude but in the opposite direction. Therefore, the magnitude of the momentum after the rebound is the same as the magnitude of the momentum just before striking the ground.
Momentum after the rebound = 2 kg x √(5gh)
Step 4: Calculate the ratio of the magnitudes of momentum.
Now, we can find the ratio of the magnitudes of momentum just before and after striking the ground.
Ratio = Momentum just before striking the ground / Momentum after the rebound
Ratio = (2 kg x √(5gh)) / (2 kg x √(5gh))
Canceling out the mass and the square roots, we get:
Ratio = 1
Therefore, the ratio of the magnitudes of momentum just before and after striking the ground is 1.