An egg with a mass m is dropped from rest from a height h and falls to the ground and breaks.
Write an expression for the magnitude of the net impulse imparted to the egg as it is stopped by the floor. Your expression will be in terms of m, h, and g. Neglect air resistance.
Expression :
J =
The correct answer and explanation is :
To find the magnitude of the net impulse imparted to the egg when it is stopped by the floor, we need to use the relationship between impulse and momentum.
Impulse and Momentum Relationship:
Impulse ( J ) is the change in momentum of an object. Mathematically, impulse is given by:
[
J = \Delta p
]
where ( \Delta p ) is the change in momentum. The momentum ( p ) of an object is defined as the product of its mass ( m ) and velocity ( v ). Therefore:
[
J = m(v_{\text{final}} – v_{\text{initial}})
]
Step 1: Initial Velocity before Impact
When the egg is dropped from height ( h ), it starts from rest. Thus, the initial velocity ( v_{\text{initial}} ) is 0. The final velocity just before it hits the ground can be found using the equation for free fall under constant acceleration due to gravity:
[
v_{\text{final}}^2 = v_{\text{initial}}^2 + 2gh
]
Since ( v_{\text{initial}} = 0 ), this simplifies to:
[
v_{\text{final}}^2 = 2gh
]
Taking the square root of both sides:
[
v_{\text{final}} = \sqrt{2gh}
]
Step 2: Impulse Calculation
The change in velocity when the egg hits the ground is:
[
\Delta v = v_{\text{final}} – 0 = \sqrt{2gh}
]
Now, we can calculate the impulse using the change in momentum:
[
J = m(v_{\text{final}} – v_{\text{initial}}) = m\sqrt{2gh}
]
Since the egg comes to a stop upon hitting the ground, the final velocity is 0, and the impulse is simply:
[
J = m\sqrt{2gh}
]
Conclusion:
The magnitude of the net impulse imparted to the egg is:
[
J = m \sqrt{2gh}
]
This expression represents the total impulse needed to stop the egg after it has fallen from height ( h ) and is subjected to the gravitational acceleration ( g ). The impulse depends on the mass of the egg, the height from which it falls, and the acceleration due to gravity.