Extreme Sports on Miranda Miranda is a satellite of Uranus, it is a roughly sphe

Question

Extreme Sports on Miranda Miranda is a satellite of Uranus, it is a roughly spherical mass of 6.68 × 1019 kg having radius 242 km.

(a) Find the free fall acceleration on its surface.
(b) There is a cliff on Miranda that is 5.00 km high. An extreme sport enthusiast runs horizontally off the

cliff at 8.50 m/s. For what time interval is he in flight? Or – is he in orbit?
(c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? (d) What is his vector impact velocity?

Explanation / Answer

(a) Use Newton’s law of gravity to find the value of g on Miranda’s surface:

F = GMm/r²

Since force is mass times gravitational acceleration (mg), you have:

mg = GMm/r²
g = GM/r²
= (6.67 x 10¹¹Nm²/kg²)(6.68 x 10¹kg) / (2.42 x 10)²
= 0.0761m/s²

(b)From what you know about the vertical motion of the jumper, the time he would fall is:
y = 0.5gt²

t = [2y / g]
= [2(5000m) / 0.0761m/s²]
= 362s

(c) The horizontal distance from the cliff’s base is:

x = (v + v)t / 2

= (8.50m/s + 8.50m/s)362s / 2

= 3077 meters

(d) His horizontal velocity is given as 8.50m/s, his vertical velocity is:

v² = v² + 2gy
= 0 + 2(-0.0761m/s²)(-5000m)
v = -27.6m/s----------------->take the minus root as velocity is downward

Then his vector velocity is:

v = [v²(x) + v²(y)]

v = [8.5^2 + 27.6^2)

v = 28.8792

Angle = arctan(-27.6/8.5)

= -72.88°

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