Question

Determine the velocity required for a moving object 5.00×10 3

m above the surface of Mars to escape from Mars's gravity. The mass of Mars is 6.42×10 23
kg, and its radius is 3.40×10 3
m.

Answer 1

The velocity required for a moving object 5.00×
`10^3` m above the surface of Mars to escape from Mars's gravity is approximately 3.66×
`10^3` m/s.

Step-by-step explanation:

To determine the escape velocity, we can use the escape velocity formula:

`\[ v = \sqrt{(2GM)/(r)} \]`

where:

`\( v \)` is the escape velocity,

`\( G \)` is the gravitational constant
`(\(6.67 * 10^(-11) \, \text{Nm}^2/\text{kg}^2\)),`

`\( M \)` is the mass of Mars
`(\(6.42 * 10^(23) \, \text{kg}\)),`

`\( r \)` is the distance from the center of Mars to the object (radius of Mars plus the height above the surface).

In this case, the radius of Mars
`(\(3.40 * 10^3 \, \text{m}\))` and the height above the surface
`(\(5.00 * 10^3 \, \text{m}\))` need to be considered. Therefore,
`\( r = 3.40 * 10^3 + 5.00 * 10^3 = 8.40 * 10^3 \, \text{m} \).`

Substituting these values into the formula:

`\[ v = \sqrt{(2 * 6.67 * 10^(-11) * 6.42 * 10^(23))/(8.40 * 10^3)} \]`

After evaluating this expression, the escape velocity
`\( v \)` is approximately
`\( 3.66 * 10^3 \, \text{m/s} \).`

This velocity represents the minimum speed an object must have to escape Mars's gravitational influence and move into space. As an object reaches this velocity, the gravitational pull from Mars is overcome, allowing the object to move freely away from the planet.

Answer 2

The velocity required for a moving object to escape Mars's gravity is approximately 5023 m/s.

Step-by-step explanation:

To determine the velocity required for a moving object to escape Mars's gravity, we can use the equation for escape velocity:

Vesc = √(2GM/r)

Where G is the universal gravitational constant (6.67 × 10-11 N m²/kg²), M is the mass of Mars (6.42 × 1023 kg), and r is the distance from the center of Mars to the object (5.00 × 103 m + 3.40 × 103 m).

Plugging in the values, we get:

Vesc = √(2 * 6.67 × 10-11 N m²/kg² * 6.42 × 1023 kg / (5.00 × 103 m + 3.40 × 103 m))

Simplifying and calculating, the escape velocity is approximately 5023 m/s.