A 50.0 kg shell was fired from a cannon at earth’s surface to a
maximum height of 400. m.
a. What is the potential energy at maximum height?
b. It then fell to a height of 100. m. What was the loss of PE as
it fell?
The Correct Answer and Explanation is :
To solve the problem, we can use the formula for gravitational potential energy (PE), which is given by:
[
PE = mgh
]
where:
- ( PE ) is the potential energy,
- ( m ) is the mass of the object (in kg),
- ( g ) is the acceleration due to gravity (approximately ( 9.81 \, \text{m/s}^2 ) near the Earth’s surface),
- ( h ) is the height above the reference point (in meters).
Part a: Potential Energy at Maximum Height
Given:
- Mass of the shell, ( m = 50.0 \, \text{kg} )
- Maximum height, ( h = 400.0 \, \text{m} )
Now, we can calculate the potential energy at the maximum height:
[
PE = mgh = (50.0 \, \text{kg}) \times (9.81 \, \text{m/s}^2) \times (400.0 \, \text{m})
]
Calculating this:
[
PE = 50.0 \times 9.81 \times 400.0 = 1962000 \, \text{J} \quad \text{(or 1.962 MJ)}
]
Part b: Loss of Potential Energy as it Fell
Next, we calculate the potential energy when the shell fell to a height of 100.0 m.
Given:
- New height, ( h = 100.0 \, \text{m} )
Now, calculate the potential energy at this height:
[
PE_{100} = mgh = (50.0 \, \text{kg}) \times (9.81 \, \text{m/s}^2) \times (100.0 \, \text{m})
]
Calculating this:
[
PE_{100} = 50.0 \times 9.81 \times 100.0 = 49050 \, \text{J}
]
Now, the loss of potential energy (ΔPE) as it fell from 400 m to 100 m is:
[
\Delta PE = PE_{400} – PE_{100} = 1962000 \, \text{J} – 49050 \, \text{J} = 1912950 \, \text{J} \quad \text{(or 1.91295 MJ)}
]
Summary
a. The potential energy at maximum height (400 m) is 1,962,000 J (or 1.962 MJ).
b. The loss of potential energy when falling from 400 m to 100 m is 1,912,950 J (or 1.91295 MJ).
Explanation
The gravitational potential energy is the energy an object possesses due to its position in a gravitational field. When the shell is fired and reaches its maximum height, all its kinetic energy is converted to potential energy. As the shell falls, it loses potential energy, which is converted back into kinetic energy. The difference in potential energy between two heights indicates how much energy has been converted during the fall, highlighting the conservation of energy principle in a gravitational field.