Question

State the law of conservation of energy? Show that when a body falls from a certain height the total mechanical energy

remains conserved.​

Answer 1

$\underline\mathfrak{What \: is \: the \: Law \: of \: Conservation \: of \: \: Energy?}$

The law of conservation of energy states that energy can neither be created nor be destroyed. Although, it may be transformed from one form to another.

Suppose a ball of mass ‘m’ falls under the effect of gravity as shown in figure.

$\underline\mathfrak\green{Show \: that \: when \: a \: body \: falls \: from \: a \: certain \: height}$ $\underline\mathfrak\green{the \: total \: mechanical \: energy \: remains \: conserved.}$

Let us find the kinetic and the potential energy of the ball at various points of its free fall. Let the ball fall from point A at a height h above the surface of the earth.

• At Point A: At point A, the ball is stationary; therefore, its velocity is zero.Therefore, kinetic energy, T = 0 and potential energy, U = mgh. Hence, total mechanical energy at point A is: E = T + U = 0 + mgh = mgh … (i)

• At Point B : Suppose the ball covers a distance x when it moves from A to B. Let v be the velocity of the ball point B. Then by the equation of motion ${v}^{2} - {u}^{2} = 2as \\ {v}^{2} - 0 = 2gh \\ or \: {v}^{2} = 2gh\\ therefore \: kinetic \: energy \: \\ t = \frac{1}{2} m {v}^{2} \\ = \frac{1}{2} \times m \times 2gh \\ mgh$And Potential energy, U = mg (h - x)Hence, total energy at point B is:E = T + U = mgx + mg(h-x) = mgh …(ii)

• At Point C : Suppose the ball covers a distance h when it moves from A to C. Let V be the velocity of the ball at point C just before it touches the ground. Then by the equation of motion and Potential energy, U = 0${v}^{2} - {u}^{2} = 2as \\ {v}^{2} - 0 = 2gh \\ or \: {v}^{2} = 2gh\\ therefore \: kinetic \: energy \: \\ t = \frac{1}{2} m {v}^{2} \\ = \frac{1}{2} \times m \times 2gh \\ mgh$Hence, total energy at point E = T + U= mgh + 0 = mgh … (iii)

Thus, it is clear from equations (i), (ii) and (iii), that the total mechanical energy of a freely falling ball remains constant.