The **mechanical energy formula**, expressed as E_{mechanical} = ½ m v^{2} + m g h, provides a way to calculate the total mechanical energy (E_{mechanical}) of an object. It combines the contributions of kinetic energy (½ m v^{2}) and potential energy (m g h). By utilizing this formula, one can determine the amount of mechanical energy possessed by an object based on its mass, velocity, gravitational acceleration, and height.

## Practice problems

### Problem #1

Calculate the mechanical energy of a 20 kg stone resting on the top of a hill at a height of 50 m. Take the value of gravitational acceleration as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Mechanical energy of a stone, E
_{mechanical}= ? - Mass of a stone, m = 20 kg
- Height at which a stone is resting, h = 50 m
- Velocity of a stone, v = 0 m/s (Because, a stone is at rest)
- Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- E
_{mechanical}= ½ m v^{2}+ m g h - E
_{mechanical}= (0) + (20 × 9.81 × 50) - E
_{mechanical}= 20 × 9.81 × 50 - E
_{mechanical}= 9,810 J

Therefore, the mechanical energy of a stone is **9,810 J**.

### Problem #2

A skydiver with a mass of 50 kg has a velocity of 20 m/s at a height of 100 m above the ground. Calculate the mechanical energy of the skydiver. Take the value of gravitational acceleration as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Mass of a skydiver, m = 50 kg
- Velocity of a skydiver, v = 20 m/s
- Height of a skydiver above the ground, h = 100 m
- Mechanical energy of a skydiver, E
_{mechanical}= ? - Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- E
_{mechanical}= ½ m v^{2}+ m g h - E
_{mechanical}= [(½ × 50 × 20^{2}) + (50 × 9.81 × 100)] - E
_{mechanical}= 10,000 + 49,050 - E
_{mechanical}= 59,050 J

Therefore, the mechanical energy of a skydiver is **59,050 J**.

### Problem #3

A 10 kg ball is rolling down a hill with a velocity of 12 m/s at a height of 30 m above the ground. Determine the mechanical energy of the ball. Take the value of gravitational acceleration as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Mass of a ball, m = 10 kg
- Velocity of a ball, v = 12 m/s
- Height of a ball above the ground, h = 30 m
- Mechanical energy of a ball, E
_{mechanical}= ? - Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- E
_{mechanical}= ½ m v^{2}+ m g h - E
_{mechanical}= [(½ × 10 × 12^{2}) + (10 × 9.81 × 30)] - E
_{mechanical}= 720 + 2943 - E
_{mechanical}= 3,663 J

Therefore, the mechanical energy of a ball is **3,663 J**.

### Problem #4

A football player kicks a football to a height of 12 m above the ground. If the mass of the football is 450 g and the velocity of the football is 8 m/s, calculate the mechanical energy of the football. Take the value of gravitational acceleration as g = 9.81 m/s^{2}.

**Solution**

Given data:

- Height of a football above the ground, h = 12 m
- Mass of a football, m = 450 g = 0.45 kg
- Velocity of a football, v = 8 m/s
- Mechanical energy of a football, E
_{mechanical}= ? - Gravitational acceleration, g = 9.81 m/s
^{2}

Applying the formula:

- E
_{mechanical}= ½ m v^{2}+ m g h - E
_{mechanical}= [(½ × 0.45 × 8^{2}) + (0.45 × 9.81 × 12)] - E
_{mechanical}= 14.4 + 52.974 - E
_{mechanical}= 67.37 J

Therefore, the mechanical energy of a football is **67.37 J**.

## Related

## More topics

- Thermal energy equation
- Potential energy formula
- Gravitational potential energy formula
- Electric potential energy formula
- Elastic potential energy formula
- Kinetic energy formula
- Rotational kinetic energy formula
- Electrical energy equation
**Mechanical energy formula**- Photon energy equation
- Conservation of energy formula

## External links

- https://study.com/skill/learn/how-to-calculate-total-mechanical-energy-explanation.html
- https://www.ck12.org/flexi/physics/conservation-of-energy/what-is-the-formula-for-total-mechanical-energy/
- https://www.studysmarter.us/explanations/physics/work-energy-and-power/total-mechanical-energy/

Deep

Learnool.com was founded by Deep Rana, who is a mechanical engineer by profession and a blogger by passion. He has a good conceptual knowledge on different educational topics and he provides the same on this website. He loves to learn something new everyday and believes that the best utilization of free time is developing a new skill.