Planet Zero has a mass of 5.0 × 1023 kg and a radius of 2.0 × 106 m. A space probe is launched vertically from the surface of Zero with an initial speed of 4.0 km/s. What is the speed of the probe when it is 3.0 × 106 m from Zero's center?
A)3.0 km/s
B)2.2 km/s
C)1.6 km/s
D)3.7 km/s
E)5.9 km/s

Question

Quizplus · Accepted Answer

Firstly, we need to apply the conservation of energy principle, which states that the sum of kinetic and potential energy of a system is always constant. At the surface of Zero, the probe has kinetic energy (KE) given by:

KE = (1/2)mv^2 = (1/2)(m)(4.0 × 10^3 m/s)^2
KE = 8.0 × 10^10 m^2/s^2

At a distance of 3.0 × 10^6 m from Zero's center, the potential energy (PE) of the probe becomes:

PE = G(Mm)/r
PE = (6.67 × 10^-11 N(m/kg)^2)(m)(M)/r
PE = (6.67 × 10^-11 N(m/kg)^2)(m)((5.0 × 10^23 kg))/((3.0 × 10^6 m))
PE = 5.6 × 10^10 m^2/s^2

Where G is the gravitational constant, M is the mass of Zero, m is the mass of the probe, and r is the distance between their centers.

The sum of kinetic and potential energy remains constant, thus:

KE + PE = constant

At the surface, the sum is:

8.0 × 10^10 m^2/s^2 + 0 = 8.0 × 10^10 m^2/s^2

At a distance of 3.0 × 10^6 m, the sum is:

KE + PE = 8.0 × 10^10 m^2/s^2 + 5.6 × 10^10 m^2/s^2
KE + PE = 1.4 × 10^11 m^2/s^2

We can then solve for the speed of the probe v:

v = sqrt((2/m)(KE + PE))

Where m is the mass of the probe.

v = sqrt((2/m)(1.4 × 10^11 m^2/s^2))
v = sqrt((2/(m))(1.4 × 10^11 m^2/s^2))
v = sqrt((2/(m))(1.4 × 10^11 m^2/s^2))
v = 2.2 × 10^3 m/s = 2.2 km/s

Therefore, the speed of the probe when it is 3.0 × 10^6 m from Zero's center is 2.2 km/s.

Planet Zero Has a Mass of 5