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1) At the top, potential energy is more
2) Halfway through the fall, potential energy is equal to kinetic energy
3) Before hitting the ground, kinetic energy is more
4) At the top: potential energy = 784 J
5) Halfway through the fall: potential energy = 392 J
6) Halfway through the fall: potential energy = 392 J
7) Before hitting the ground: kinetic energy = 784 J
Explanation:
1)
The gravitational potential energy (GPE) of an object is the energy possessed by the object due to its position in a gravitational field. It is calculated as
where:
m is the mass of the object
is the acceleration of gravity
h is the height of the object relative to the ground
Instead, the kinetic energy of an object is the energy possessed by the object due to its motion; it is calculated as
where
v is the speed of the object
In this problem, when the bowling ball sits on top of the building its speed is zero:
v = 0
Therefore its kinetic energy is zero: KE = 0.
Instead, its potential energy is
where H is the height of the building: therefore, potential energy is more than kinetic energy.
2)
According to the law of conservation of energy, the total mechanical energy of the ball E (sum of potential and kinetic energy) is constant during the fall:
Where GPE is the potential energy and KE the kinetic energy.
When the ball is at the top of the building, it has only potential energy, as the kinetic energy is zero, therefore:
When the ball is halfway through the fall, the height is instead H/2, so:
Therefore the potential energy is now half of the total mechanical energy: and since the total energy must be constant, the kinetic must be also equal to half of the total energy, E/2.
Therefore potential and kinetic energy are equal.
3)
Just before hitting the ground, the height of the ball has became zero, so now the potential energy is zero:
This means that now all the mechanical energy of the ball is kinetic energy:
And therefore, the kinetic energy is now more than the potential energy. This is due to the fact that as the balls falls down, it accelerates, so its speed increases, and therefore potential energy is converted into kinetic energy during the fall.
4)
The gravitational potential energy of the ball is given by the formula
where:
m is the mass of the ball
g is the acceleration of gravity
h is the height of the ball above the ground
When the ball is at the top of the building, we have:
m = 2 kg
H = 40 m (height of the building)
And so, the potential energy is:
5)
Here potential energy of the ball is given by:
where:
m = 2 kg is the mass
is the acceleration of gravity
And the height of the ball is now
h = 20 m
Since it is halfway through the fall.
Therefore, the gravitational potential energy is
So, the potential energy is now half of the initial potential energy, since the other half has converted into kinetic energy.
6)
The kinetic energy of the ball is given by:
where:
m is the mass of the ball
v is its speed
Here the ball is halfway through the fall, so we have:
m = 2 kg (mass of the ball)
v = 19.8 m/s (speed)
And so, the kinetic energy is
Therefore, the kinetic energy is equal to the potential energy when the ball is halfway through the fall.
7)
The kinetic energy of the ball before hitting the ground is
where
m = 2 kg is the mass of the ball
v = 28 m/s is the speed
So, kinetic energy is
This value is equal to the value of the potential energy when the ball was at the top of the building: this means that during the fall, all the initial potential energy has been converted into kinetic energy.
Learn more about kinetic and potential energy:
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