Explain what is meant by escape speed.

Titania is a moon that orbits the planet Uranus. The mass of Titania is 3.5×10²¹kg. The radius of Titania is 800 km.

(i) Use the data to calculate the gravitational potential at the surface of Titania.

(ii) Use your answer to (b)(i) to determine the escape speed for Titania.

An astronaut visiting Titania throws an object away from him with an initial horizontal velocity of 1.8 m s^–1. The object is 1.5 m above the moon’s surface when it is thrown. The gravitational field strength at the surface of Titania is 0.37 N kg^–1.

Calculate the distance from the astronaut at which the object first strikes the surface.

(minimum) speed of object to escape gravitational field of a planet/travel to infinity;
at surface of planet;
without (further) energy input;

(i) \( - \frac{{6.67 \times {{10}^{ - 11}} \times 3.5 \times {{10}^{21}}}}{{8.0 \times {{10}^5}}}\);
−2.9×10⁵Jkg^–1; (allow Nmkg⁻¹)
Award [1 max] if negative sign omitted.

(ii) \(\frac{1}{2}m{v^2} = mV\);
speed=\( = \sqrt {2 \times 2.9 \times {{10}^5}} \); (allow ECF from (b)(i))
7.6 ×10²ms^-1;
Ignore sign.
Award [3] for a bald correct answer.

time to hit surface \( = \sqrt {\frac{{2.0 \times 1.5}}{{0.37}}} \left( { = 2.85{\rm{s}}} \right)\);
distance to impact = 2.85×1.8;
5.1m;