Two people are standing in the middle of a frozen pond that is frictionless. They want to move to the edges of the pond but are
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1) At the top of the building, the ball has more potential energy
2) When the ball is halfway through the fall, the potential energy and the kinetic energy are equal
3) Before hitting the ground, the ball has more kinetic energy
4) The potential energy at the top of the building is 784 J
5) The potential energy halfway through the fall is 392 J
6) The kinetic energy halfway through the fall is 392 J
7) The kinetic energy just before hitting the ground is 784 J
Explanation:
1)
The potential energy of an object is given by
where
m is the mass
g is the acceleration of gravity
h is the height relative to the ground
While the kinetic energy is given by
where v is the speed of the object
When the ball is sitting on the top of the building, we have
, therefore the potential energy is not zero
, since the ball is at rest, therefore the kinetic energy is zero
This means that the ball has more potential energy than kinetic energy.
2)
When the ball is halfway through the fall, the height is
So, half of its initial height. This also means that the potential energy is now half of the potential energy at the top (because potential energy is directly proportional to the height).
The total mechanical energy of the ball, which is conserved, is the sum of potential and kinetic energy:
At the top of the building,
While halfway through the fall,
And the mechanical energy is
which means
So, when the ball is halfway through the fall, the potential energy and the kinetic energy are equal, and they are both half of the total energy.
3)
Just before the ball hits the ground, the situation is the following:
- The height of the ball relative to the ground is now zero:
. This means that the potential energy of the ball is zero:
- The kinetic energy, instead, is not zero: in fact, the ball has gained speed during the fall, so
, and therefore the kinetic energy is not zero
Therefore, just before the ball hits the ground, it has more kinetic energy than potential energy.
4)
The potential energy of the ball as it sits on top of the building is given by
where:
m = 2 kg is the mass of the ball
is the acceleration of gravity
h = 40 m is the height of the building, where the ball is located
Substituting the values, we find the potential energy of the ball at the top of the building:
5)
The potential energy of the ball as it is halfway through the fall is given by
where:
m = 2 kg is the mass of the ball
is the acceleration of gravity
h = 20 m is the height of the ball relative to the ground
Substituting the values, we find the potential energy of the ball halfway through the fall:
6)
The kinetic energy of the ball halfway through the fall is given by
where
m = 2 kg is the mass of the ball
v = 19.8 m/s is the speed of the ball when it is halfway through the fall
Substituting the values into the equation, we find the kinetic energy of the ball when it is halfway through the fall:
We notice that halfway through the fall, half of the initial potential energy has converted into kinetic energy.
7)
The kinetic energy of the ball just before hitting the ground is given by
where:
m = 2 kg is the mass of the ball
v = 28 m/s is the speed of the ball just before hitting the ground
Substituting the values into the equation, we find the kinetic energy of the ball just before hitting the ground:
We notice that when the ball is about to hit the ground, all the potential energy has converted into kinetic energy.
Learn more about kinetic and potential energy:
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