Zero, a hypothetical planet, has a mass of 2.7 x 10^23 kg, a radius of 2.8 x 10^6 m, and no atmosphere. A 3.6 kg space probe is to be launched vertically from its surface. (b) If the probe is to achieve a maximum distance of 4.6 × 10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

Question

Zero, a hypothetical planet, has a mass of 2.7 x 10^23 kg, a radius of 2.8 x 10^6 m, and no atmosphere. A 3.6 kg space probe is to be launched vertically from its surface. (b) If the probe is to achieve a maximum distance of 4.6 × 10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?

Answer 1

The kinetic energy when it reaches its final height must be 0. By conservation of energy, the energy it has at its final height is equal to the energy when it is launched.

Answer 2

Got it & Thanks! I was just messing up the (-) signs the whole time. Thanks.

Answer 3

To calculate the initial kinetic energy required to launch the space probe from the surface of Zero to achieve a maximum distance of 4.6 × 10^6 m from its center, we can use the principle of conservation of energy.

The total mechanical energy of the probe at the surface of Zero will be equal to the sum of its gravitational potential energy and its initial kinetic energy.

1. Calculate the gravitational potential energy:
The gravitational potential energy (U) is given by the formula: U = -GMm/r
where G is the gravitational constant (6.67430 × 10^-11 N m²/kg²), M is the mass of Zero (2.7 × 10^23 kg), m is the mass of the probe (3.6 kg), and r is the radius at which the probe achieves the maximum distance (4.6 × 10^6 m).

Substituting these values into the formula:
U = -[(6.67430 × 10^-11 N m²/kg²) * (2.7 × 10^23 kg) * (3.6 kg)] / (4.6 × 10^6 m)

2. Calculate the initial kinetic energy:
The initial kinetic energy (K) can be calculated using the equation: K = total mechanical energy - gravitational potential energy
Since the maximum distance is given, the total mechanical energy is equal to zero (as the probe will come to a stop at 4.6 × 10^6 m from the center of Zero).

Substituting the values into the equation:
K = 0 - U

Now you can calculate the initial kinetic energy of the probe by substituting the value of U into the equation.

Note: The value of U will be negative because it represents the gravitational potential energy at the surface of Zero.