Question

HELP PLEASE!!
You throw a rock with sufficient speed to put it into orbit around the asteroid 234 Ida very close to its surface. How long would it take the rock to make one orbit around the asteroid? Assume 234 Ida is spherical. Ida's mass is 4 x 1016 kg and its radius is 16 km.

A) 0.62 hr
B) 2.2 hr
C) 1.6 hr
D) 3.4 hr

Answer 1

Answer:

The correct option is;

B) 2.2 hr

Explanation:

The location of the rock in the orbit of the asteroid = Close to the surface of the asteroid

The required equation is given as follows;

$\dfrac{G \times M \times m}{r^2} = \dfrac{m \times v^2}{r}$

$\therefore v^2 = \dfrac{G \times M}{r}$

Where;

v = The orbital velocity

G = The universal gravitational constant = 6.67430 × 10⁻¹¹ N·m²/kg²

r = The radius of the planet = 16 km = 16,000 m

M = The mass of the planet = 4 × 10¹⁶ kg

∴ v² = 6.67430 × 10⁻¹¹ × 4 × 10¹⁶/16,000 = 166.8575

v = √(166.8575) = 12.92 m/s

The orbital velocity = v = 12.92 m/s

The time it takes for the rock to make one orbit round the asteroid is given as follows

The length of the orbit = The circumference of the asteroid = 2 × π × r  

The length of the orbit =  2 × π × 16,000 ≈ 100530.965 m

The time it takes for the rock to make one orbit round the asteroid = The length of the orbit/(The speed of the asteroid)

The time it takes for the rock to make one orbit round the asteroid = 100530.965/(12.92) ≈ 7781.034 seconds or 2.2 hours (by approximation by one decimal place)

The time it takes for the rock to make one orbit round the asteroid ≈ 2.2 hours.