# Mechanical energy formula

The mechanical energy formula, expressed as Emechanical = ½ m v2 + m g h, provides a way to calculate the total mechanical energy (Emechanical) of an object. It combines the contributions of kinetic energy (½ m v2) 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.

Contents

## 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/s2.

Solution

Given data:

• Mechanical energy of a stone, Emechanical = ?
• 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/s2

Applying the formula:

• Emechanical = ½ m v2 + m g h
• Emechanical = (0) + (20 × 9.81 × 50)
• Emechanical = 20 × 9.81 × 50
• Emechanical = 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/s2.

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, Emechanical = ?
• Gravitational acceleration, g = 9.81 m/s2

Applying the formula:

• Emechanical = ½ m v2 + m g h
• Emechanical = [(½ × 50 × 202) + (50 × 9.81 × 100)]
• Emechanical = 10,000 + 49,050
• Emechanical = 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/s2.

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, Emechanical = ?
• Gravitational acceleration, g = 9.81 m/s2

Applying the formula:

• Emechanical = ½ m v2 + m g h
• Emechanical = [(½ × 10 × 122) + (10 × 9.81 × 30)]
• Emechanical = 720 + 2943
• Emechanical = 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/s2.

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, Emechanical = ?
• Gravitational acceleration, g = 9.81 m/s2

Applying the formula:

• Emechanical = ½ m v2 + m g h
• Emechanical = [(½ × 0.45 × 82) + (0.45 × 9.81 × 12)]
• Emechanical = 14.4 + 52.974
• Emechanical = 67.37 J

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