# Application of the conservation of mechanical energy for vertical motion in free fall – problems and solutions

1. A person throws a 1-kg stone upward at 2 m/s while standing on the edge of a cliff so that the stone can fall to the base of the cliff 40 meters below. What is the kinetic energy of stone at 10 meters above the ground? Acceleration due to gravity g = 10 m/s^{2}.

__Known :__

Mass (m) = 1 kg

Initial velocity (v_{o}) = 2 m/s

The change in height = 40 – 10 = 30 meters

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted :__ kinetic energy of stone at 10 meters above the ground

__Solution :__

**The initial mechanical energy**

The initial gravitational potential energy** **(EP) = m g h = (1)(10)(30) = 300 Joule

The initial kinetic energy (KE) = ½ m v_{o}^{2} = ½ (1)(2)^{2} = ½ (4) = 2 Joule

The initial mechanical energy = the initial gravitational potential energy + the initial kinetic energy = 300 + 2 = 302 Joule.

**The final mechanical energy **

The final mechanical energy = the final kinetic energy = the initial gravitational potential energy + the initial kinetic energy = 300 + 2 = 302 Joule.

2. A person throws a 1-kg object upward into the air with an initial velocity of 10 m/s. Determine (a) the gravitational potential energy at the maximum height (b) the maximum height.

Acceleration due to gravity is 10 m/s^{2}

__Known :__

Mass (m) = 1 kg

Initial velocity (v_{o}) = 10 m/s

Acceleration due to gravity (g) = 10 m/s^{2}

__Wanted __: the gravitational potential energy at the maximum height and the maximum height

__Solution :__

__(a) the gravitational potential energy at the maximum height__

**The initial mechanical energy :**

The initial mechanical energy (ME) = the initial kinetic energy (KE) = ½ m v_{o}^{2} = ½ (1)(10)^{2} = ½ (100) = 50 Joule.

**The final mechanical energy : **

The final mechanical energy (ME) = the gravitational potential energy (PE)

**The principle of conservation of mechanical energy :**

The initial mechanical energy = the final mechanical energy

KE = PE

50 = PE

The gravitational potential energy is 50 Joule.

__(b) The maximum height__

PE = m g h

50 = (1)(10) h

50 = 10 h

h = 50 / 10 = 5 meters

The maximum height is 5 meters above the ground.

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