Suppose the New Horizons spacecraft comes within 25,000 km of the dwarf planet in Question 1, which is 2 × greater than its closest approach to Pluto. If it feels a gravitational force that is 1/2 as strong as the gravitational force it felt at its closest approach to Pluto, what is the mass of this dwarf planet? Show how you got your answer.
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We can solve this by using the Newton's Law of universal gravitation, according to which the force of gravity is directly proportional to the product of masses of the two bodies and is inversely proportional to the square of the distance between them.
or, `F = G (m_1m_2)/r^2`
In this case, if m1 is the mass of the planet and m2 is the mass of the spacecraft and r is the distance between them, the gravitational force between the dwarf and spacecraft can be given as,
Fd = G m1m2/r^2
It is also given that in case of Pluto, spacecraft is at half the distance (i.e, r/2) and it experiences half gravitational force from Dwarf planet as compared to Pluto, i.e. Fp = 2Fd
and assuming mass of pluto as mp, we get
Fg = Gmpm2/(r/2)^2 = 4 Gmpm2/r^2 = 2Fd = 2 Gm1m2/(r^2)
or, m1 = 2 mp
i.e. the dwarf planet has 2 times the mass of Pluto.
Hope this helps.
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