Zero, a hypothetical planet, has a mass of 4.1 x 1023 kg, a radius of 3.0 x 106 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

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Asked Nov 17, 2019
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Zero, a hypothetical planet, has a mass of 4.1 x 1023 kg, a radius of 3.0 x 106 m, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 5.0 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

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Expert Answer

Step 1

Write the formula for the gravitational potential energy.

-GM
U
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-GM U

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Step 2

(a)

Consider r be the distance from the center to the surface and R be the distance from the center to the final position.

From conservation of energy.

KE,+U, KE, +U,
GM
KE, -
GMm
zero
-ΚΕ,
R
1
KE, KE,+GMe
1
aro
rR
GM (-Rr
KE, KE,
zero
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KE,+U, KE, +U, GM KE, - GMm zero -ΚΕ, R 1 KE, KE,+GMe 1 aro rR GM (-Rr KE, KE, zero

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Step 3

Substitute the known val...

(6.7x101 N-m /kg)(4.1x10 kg)x
(10 kg)(-4.0x10 m+3.0x10 m)
KE, 5.0x10 J+
(3.0x10 m)(4.0x10 m)
2.7x10
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(6.7x101 N-m /kg)(4.1x10 kg)x (10 kg)(-4.0x10 m+3.0x10 m) KE, 5.0x10 J+ (3.0x10 m)(4.0x10 m) 2.7x10

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