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

To find the speed of the probe at a certain distance from the planet's center, we can use the conservation of mechanical energy principle. The total mechanical energy (kinetic + potential) at the surface must equal the total mechanical energy at the new distance. The formula for gravitational potential energy is -GMm/r, where G is the gravitational constant, M is the mass of the planet, m is the mass of the probe (which cancels out in this calculation), and r is the distance from the center of the planet. The kinetic energy is given by 1/2 mv^2, where v is the velocity of the probe.Given:- Initial distance from center (r1) = radius of Zero = 2.0 × 10^16 m- Final distance from center (r2) = 3.0 × 10^16 m- Mass of Zero (M) = 5.0 × 10^23 kg- Initial speed (v1) = 4.0 km/s = 4000 m/sThe initial potential energy (U1) = -GMm/r1, and the final potential energy (U2) = -GMm/r2. The initial kinetic energy (K1) = 1/2 mv1^2, and we need to find the final speed v2, so the final kinetic energy (K2) = 1/2 mv2^2.By conservation of energy: K1 + U1 = K2 + U2Solving for v2 involves substituting the known values and solving the equation, which results in v2 ≈ 2.2 km/s. This calculation requires knowledge of G, the gravitational constant, but since the exact calculation steps are not shown here, the key point is understanding that energy conservation allows us to solve for the final speed of the probe.

Planet Zero Has a Mass of 5