An animal-rescue plane flying due east at 33.0 m/s drops a bale of hay from an altitude of 58.0 m. If the bale of hay weighs 173 N, what is the momentum of the bale the moment it strikes the ground?
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To calculate the momentum of the bale of hay just before it strikes the ground, we need to find its velocity.
We know that the plane is flying due east at a speed of 33.0 m/s, but we don't know how fast the bale of hay is falling.
To find the velocity of the falling bale, we can use the kinematic equation:
v^2 = u^2 + 2as
Where:
v = final velocity of the bale (which we want to find)
u = initial velocity of the bale (0 m/s since it was dropped from rest)
a = acceleration of the bale due to gravity (-9.8 m/s^2, since it is falling vertically downwards)
s = displacement of the bale (58.0 m, its initial altitude)
Plugging the values into the equation, we get:
v^2 = 0^2 + 2(-9.8)(58)
v^2 = 2(-9.8)(58)
v^2 = -1137.6
To solve for v, we take the square root of both sides:
v = √(-1137.6)
v ≈ -33.73 m/s
Since velocity is a vector quantity, the negative sign indicates that the bale is moving downwards.
Lastly, to calculate the momentum of the bale just before it strikes the ground, we use the formula:
momentum = mass × velocity
We are given the weight of the bale, which is the force exerted on the bale due to gravity:
weight = mass × acceleration due to gravity
Since weight equals mass times acceleration due to gravity, we can rearrange the formula to find the mass:
mass = weight / acceleration due to gravity
Substituting the given values, we have:
mass = 173 N / 9.8 m/s^2
mass ≈ 17.7 kg
Now, we can calculate the momentum:
momentum = mass × velocity
momentum = 17.7 kg × -33.73 m/s
momentum ≈ -595.10 kg·m/s
Therefore, the momentum of the bale the moment it strikes the ground is approximately -595.10 kg·m/s.