Escape!

If an object is thrown vertically upwards at speed $11.2\text{ km/s},$ it escapes from the surface of Earth. What will be the escape speed for the same object if it is thrown at an angle of $30^\circ$ with the vertical?

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Details and Assumptions:

• Air resistance is neglected.
• The object escapes when its total mechanical energy that is the sum of kinetic energy $\big(\frac{1}{2} mv^2\big)$ and gravitational potential energy $\left(-\frac{GMm}{R^2}\right)$ is zero, where $m$ is the mass of the object, $M$ is the mass of Earth, and $v$ is the speed of projection of the object.
• Neglect the effects due to rotation and shape of the Earth, assume it is thrown from one of the poles.