Answer :
To find the total mechanical energy of the plane, we need to consider both its potential energy and its kinetic energy.
### Step 1: Calculate Potential Energy
Potential energy (PE) is the energy stored due to the plane's height off the ground. The formula to calculate potential energy is:
[tex]\[ \text{PE} = m \times g \times h \][/tex]
where:
- [tex]\( m = 2000 \, \text{kg} \)[/tex] is the mass of the plane,
- [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex] is the acceleration due to gravity,
- [tex]\( h = 250 \, \text{m} \)[/tex] is the height of the plane.
By plugging the given values into the formula, we find:
[tex]\[ \text{PE} = 2000 \times 9.81 \times 250 = 4,905,000 \, \text{Joules} \][/tex]
### Step 2: Calculate Kinetic Energy
Kinetic energy (KE) is the energy due to the plane's motion. The formula to calculate kinetic energy is:
[tex]\[ \text{KE} = \frac{1}{2} \times m \times v^2 \][/tex]
where:
- [tex]\( m = 2000 \, \text{kg} \)[/tex] is the mass of the plane,
- [tex]\( v = 122 \, \text{m/s} \)[/tex] is the speed of the plane.
By substituting the given values into the formula, we get:
[tex]\[ \text{KE} = \frac{1}{2} \times 2000 \times (122)^2 = 14,884,000 \, \text{Joules} \][/tex]
### Step 3: Total Mechanical Energy
The total mechanical energy (TME) of the plane is the sum of its potential and kinetic energies:
[tex]\[ \text{TME} = \text{PE} + \text{KE} \][/tex]
Substitute the calculated values:
[tex]\[ \text{TME} = 4,905,000 + 14,884,000 = 19,789,000 \, \text{Joules} \][/tex]
Thus, the total mechanical energy of the plane is 19,789,000 Joules.
### Step 1: Calculate Potential Energy
Potential energy (PE) is the energy stored due to the plane's height off the ground. The formula to calculate potential energy is:
[tex]\[ \text{PE} = m \times g \times h \][/tex]
where:
- [tex]\( m = 2000 \, \text{kg} \)[/tex] is the mass of the plane,
- [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex] is the acceleration due to gravity,
- [tex]\( h = 250 \, \text{m} \)[/tex] is the height of the plane.
By plugging the given values into the formula, we find:
[tex]\[ \text{PE} = 2000 \times 9.81 \times 250 = 4,905,000 \, \text{Joules} \][/tex]
### Step 2: Calculate Kinetic Energy
Kinetic energy (KE) is the energy due to the plane's motion. The formula to calculate kinetic energy is:
[tex]\[ \text{KE} = \frac{1}{2} \times m \times v^2 \][/tex]
where:
- [tex]\( m = 2000 \, \text{kg} \)[/tex] is the mass of the plane,
- [tex]\( v = 122 \, \text{m/s} \)[/tex] is the speed of the plane.
By substituting the given values into the formula, we get:
[tex]\[ \text{KE} = \frac{1}{2} \times 2000 \times (122)^2 = 14,884,000 \, \text{Joules} \][/tex]
### Step 3: Total Mechanical Energy
The total mechanical energy (TME) of the plane is the sum of its potential and kinetic energies:
[tex]\[ \text{TME} = \text{PE} + \text{KE} \][/tex]
Substitute the calculated values:
[tex]\[ \text{TME} = 4,905,000 + 14,884,000 = 19,789,000 \, \text{Joules} \][/tex]
Thus, the total mechanical energy of the plane is 19,789,000 Joules.
