You just landed on the planet Olympus IV, famous for its sports competitions. It's a light planet: the average density is one quarter of the Earths average density. It's also a large planet: the diameter is the double of the Earths diameter. The Olympians are giving a high jump competition, and one crew member takes part. You know that on Earth he jumps 1.98 meters. The record at Olympus IV is 3.86 meters. Should you bet your salary on him winning the competition ?
(In reply to
Answer by K Sengupta)
Let us denote:
D_L = average density of Olympus IV
D_E = average density of Earth
G_L = gravitation of Olympus IV
G_E = gravitation of Earth
m_L = mass of Olympus IV
m_E = mass of Earth
R_L = radius of Olympus IV
R_E = radius of Earth
V_L = volume of Olympus IV
V_E = volume of Earth
M = mass of the athlete
It is given that:
(i) R_L = 2*R_E, and (ii) D_L = 0,25*D_E
Now, we know that:
Average density varies directly(vd) as the volume.
Thus, V_L vd R_L^3, and:
V_L vd r_E^3, so that from (ii), we obtain:
V_L/V_E = 8
Now, m = D*V
-> m_L/m_E = 8*o,25 = 2
-> m_L = 2*m_E
Then,
G_L/G_E
= (m_L*M)/(R_l^2)*(m_E*m/R_E^2)^-1
= 2/4
= 0.5
Let the respective height to which the crew member can jump on Olympus
IV and Earth be h_L and h_E.
Let c be the gravitational center.
Then,
g_L/g_E = (h_L - c)/(h_E - c)
Now, we know that c = 1 meter and it is given that: h_E = 1.98 meters
Thus,
h_L = 2(1.98 - 1)
-> h_L = 2.96 meters.
Consequently, it would be unwise to bet one's salary upon the crew member winning the competition.
Edited on December 15, 2008, 6:41 am
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