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A projectile is launched with speed v = sqrt(GM/r) at an angle theta (where theta is not 0°, 90°, or 180°) with respect to the local radius vector at a distance r from Earth's centre (M = mass of Earth). Which of the following statements is INCORRECT?

1. The projectile follows an elliptical orbit.
2. The launch point lies at the end of the semi-minor axis of the elliptical orbit.
3. If the projectile is initially moving towards Earth, then Earth is located at the nearer focus of the ellipse.
4. Earth can be located at either focus of the ellipse depending on initial conditions.

Correct answer: Earth can be located at either focus of the ellipse depending on initial conditions.

Solution

With v = sqrt(GM/r) and vis-viva equation, the semi-major axis a = r, placing the launch point at the end of the semi-minor axis. Earth is at a specific focus determined by whether the projectile is approaching or receding — it cannot be at either focus arbitrarily. Statement D is incorrect.

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