A satellite moves around the earth in a circular orbit with speed v. If m is the mass of the satellite, its total energy is

The correct option is -mv²/2.

Let us derive the expression for the total energy of a satellite of mass $m$ orbiting the Earth (of mass $M$) in a circular orbit of radius $r$ with a constant speed $v$.

First, the kinetic energy ($E_k$) of the satellite is given by:
$E_k=\frac{1}{2}m{v}^2$

Next, we determine the gravitational potential energy ($E_p$). For a satellite in a stable circular orbit, the centripetal force is provided entirely by the gravitational force between the Earth and the satellite:
$\frac{m{v}^2}{r}=\frac{GMm}{r^2}$
where $G$ is the universal gravitational constant.

Simplifying the equation above by multiplying both sides by $r$ gives:
$m{v}^2=\frac{GMm}{r}$

The gravitational potential energy of the system is defined as:
$E_p=-\frac{GMm}{r}$

Substituting $m{v}^2$ into the potential energy expression, we get:
$E_p=-m{v}^2$

The total energy ($E$) of the satellite is the sum of its kinetic energy and potential energy:
$E=E_k+E_p$
$E=\frac{1}{2}m{v}^2+(-m{v}^2)$
$E=-\frac{1}{2}m{v}^2=-m{v}^2/2$

Thus, the total energy of the satellite is indeed negative, indicating that it is in a bound state orbiting the Earth, with a value of -mv²/2.