Law of conservation of mechanical energy

Total mechanical energy, the conversion of kinetic energy into potential energy and back, the law of conservation of mechanical energy when there is no friction.

Terms

Studied 0 of 3

Formulas

Total mechanical energy

E = E_к + E_п

Quantities
Symbol	Name	Unit
$E$	total mechanical energy	J
$E_к$	kinetic energy	J
$E_п$	potential energy	J

Solved for another quantity:

E_к = E - E_п

E_п = E - E_к

Note. For example, a $2\ \text{kg}$ body at a height of $5\ \text{m}$ moves at $10\ \text{m/s}$ ( $g = 10$ ): $E_k = 100\ \text{J}$ , $E_p = 100\ \text{J}$ , $E = 200\ \text{J}$ .

Law of conservation of mechanical energy

E_{к1} + E_{п1} = E_{к2} + E_{п2} = \text{const}

Quantities
Symbol	Name	Unit
$E_{к1},\ E_{п1}$	kinetic and potential energy in the first state	J
$E_{к2},\ E_{п2}$	kinetic and potential energy in the second state	J

Solved for another quantity:

E_{к2} = E_{к1} + E_{п1} - E_{п2}

E_{п2} = E_{к1} + E_{п1} - E_{к2}

Note. True when there is no friction. For example, a body is dropped from rest from a height of $20\ \text{m}$ ( $g = 10$ ): $m g h = \dfrac{m v^2}{2}$ , so at the ground $v = \sqrt{2 g h} = \sqrt{2\cdot 10\cdot 20} = 20\ \text{m/s}$ .

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Flight of a thrown body

The column on the right is the total energy E: the blue part is potential, the yellow part is kinetic. The parts change, the height of the whole column does not.

t

= 0 sPlayback speed

Graphs

E_k(t),\ E_p(t),\ E(t)

Yellow is E_k, blue is E_p, grey is their sum E: a horizontal line

E_k

100 J

E_p

200 J

E

300 J

v

10 m/s

Formulas with your numbers

E_{k} = \dfrac{m v^2}{2} = \dfrac{2\cdot 10^2}{2} = 100\ \text{J}

E_{p} = m g h = 2\cdot 10\cdot 10 = 200\ \text{J}

E = E_{k} + E_{p} = 100 + 200 = 300\ \text{J}

The flight lasts 2.73 s, the greatest height is 15 m.

How to read the graphs

On the $E_k(t)$ and $E_p(t)$ graphs the curves mirror each other: when one energy decreases, the other increases by the same amount.
The graph of the total energy $E(t)$ is a horizontal line: the sum $E_k + E_p$ is the same at every moment (there is no friction).
At the top point the speed is $0$ , so $E_k = 0$ and $E_p$ is the greatest. Right at the ground $E_p = 0$ and $E_k$ is the greatest and equals $E$ .
The greater the initial speed or height, the higher the horizontal line $E(t)$ . The mass changes the scale of all three graphs, but not their shape.

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Terms

Studied 0 of 3

Formulas

Total mechanical energy

E = E_к + E_п

Quantities
Symbol	Name	Unit
$E$	total mechanical energy	J
$E_к$	kinetic energy	J
$E_п$	potential energy	J

Solved for another quantity:

E_к = E - E_п

E_п = E - E_к

Note. For example, a $2\ \text{kg}$ body at a height of $5\ \text{m}$ moves at $10\ \text{m/s}$ ( $g = 10$ ): $E_k = 100\ \text{J}$ , $E_p = 100\ \text{J}$ , $E = 200\ \text{J}$ .

Law of conservation of mechanical energy

E_{к1} + E_{п1} = E_{к2} + E_{п2} = \text{const}

Quantities
Symbol	Name	Unit
$E_{к1},\ E_{п1}$	kinetic and potential energy in the first state	J
$E_{к2},\ E_{п2}$	kinetic and potential energy in the second state	J

Solved for another quantity:

E_{к2} = E_{к1} + E_{п1} - E_{п2}

E_{п2} = E_{к1} + E_{п1} - E_{к2}

Lab

Set the quantities with the sliders. The graphs and the formulas with your numbers update at once.

Flight of a thrown body

The column on the right is the total energy E: the blue part is potential, the yellow part is kinetic. The parts change, the height of the whole column does not.

t

= 0 sPlayback speed

Graphs

E_k(t),\ E_p(t),\ E(t)

Yellow is E_k, blue is E_p, grey is their sum E: a horizontal line

E_k

100 J

E_p

200 J

E

300 J

v

10 m/s

Formulas with your numbers

E_{k} = \dfrac{m v^2}{2} = \dfrac{2\cdot 10^2}{2} = 100\ \text{J}

E_{p} = m g h = 2\cdot 10\cdot 10 = 200\ \text{J}

E = E_{k} + E_{p} = 100 + 200 = 300\ \text{J}

The flight lasts 2.73 s, the greatest height is 15 m.

How to read the graphs

On the

E_k(t)

and

E_p(t)

graphs the curves mirror each other: when one energy decreases, the other increases by the same amount.

The graph of the total energy

E(t)

is a horizontal line: the sum

E_k + E_p

is the same at every moment (there is no friction).

At the top point the speed is

0

, so

E_k = 0

and

E_p

is the greatest. Right at the ground

E_p = 0

and

E_k

is the greatest and equals

E

The greater the initial speed or height, the higher the horizontal line

E(t)

. The mass changes the scale of all three graphs, but not their shape.