A projectile launched very fast

Adapted from IOAA 2009 (Tehran), theory problem 16. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

A projectile leaves Earth's surface at sea level with speed $v_0=\sqrt{GM/R}$, at an angle $\theta =30°$ above the local horizontal, where $M$ and $R$ are Earth's mass and radius. Ignore the air and Earth's rotation.

(a) Show that the path is an ellipse with semi-major axis $a=R$.

(b) What is the greatest height of the projectile above the surface, in Earth radii?

(c) What is its range, the distance along the surface from launch to landing, in Earth radii?

(d) What is the eccentricity of the ellipse?

(e) How long is the projectile in flight?

Answer(a) $a=R$. (b) $0.5R$. (c) $2\pi R/3=2.094R$. (d) $e=0.5$. (e) $(\pi +1)\sqrt{R^3/GM}$, which is 55.6 minutes, or 0.659 of the period of an orbit with $a=R$. The launch figure in Section 3.3 shows this path.