Question

A projectile is fired with a speed of $\sqrt{\frac{GM}{r}}$ at certain angle ($\theta \ne 0^\circ, 90^\circ, 180^\circ$) with the radius vector. Here r is distance of projectile from center of Earth and M is mass of Earth. Choose the INCORRECT statement.

• A. The projectile will follow elliptical path.
• B. The projectile is at semi-minor axis of its elliptical path.
• C. Earth is at nearer focus if the projectile is initially approaching the Earth.
• D. Earth could be at any focus of the ellipse.

Answer 1

Answer: (C)

Earth is at nearer focus if the projectile is initially approaching the Earth.

Answer 2

The total mechanical energy $E$ of the projectile is given by $E = K + U = \frac{1}{2}mv^2 - \frac{GMm}{r}$. Substituting the given speed $v = \sqrt{\frac{GM}{r}}$, we get $E = -\frac{GMm}{2r}$. Since $E < 0$, the path is elliptical.

For an elliptical orbit, $E = -\frac{GMm}{2a}$, where $a$ is the semi-major axis. Comparing this with the calculated energy, we find $a = r$.

The distance from a focus to the ends of the semi-minor axis of an ellipse is equal to the semi-major axis $a$. Since the projectile's initial distance from Earth (at a focus) is $r$, and $r = a$, the projectile is initially at an end of the semi-minor axis.

An ellipse has two foci. Earth is at one focus ($F_1$). The projectile is at P. The distance from Earth is $r$. The distance from the other focus ($F_2$) is $2a - r$.

• If $r < a$, Earth is at the nearer focus ($r < 2a - r$).
• If $r > a$, Earth is at the farther focus ($r > 2a - r$).
• If $r = a$, Earth is at an equidistant focus ($r = 2a - r$).

Since the initial position is at $r = a$, the projectile is equidistant from both foci. Therefore, the statement "Earth is at nearer focus if the projectile is initially approaching the Earth" is incorrect for the initial moment.

By definition, the central body (Earth) occupies one of the two foci of the elliptical orbit.