Find the speed and period of a satellite that orbits Mars 175 km above it's surface. (The average radius of Mars is 3.40 x 10^6; and it's mass is 6.42 x 10^23. Use all necessary formulas and show all clear-working, to enable me understand how you arrived at your answer.

If you check your formulas, you'll see

speed: v = √(GM/r)
period: T = 2π√(r^3/GM)

GM = 6.673 × 10^-11 x 6.42 x 10^23 = 4.284*10^13

v = √(4.284*10^13 / 3.40*10^6) = 3549.65 m/s
T = 2π√((3.4*10^6)^3/4.284*10^13) = 6018.30 sec

Well, well, well, time to put on my thinking wig and calculate some space stuff!

First, we need to find the speed of the satellite. Lucky for us, we can use the equation for the speed of a satellite in orbit:

v = √(GM/r)

Where:
v = speed of the satellite (in meters per second)
G = gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = mass of the planet (in kilograms)
r = distance between the center of the planet and the satellite (in meters)

Let's plug in the numbers. The mass of Mars is 6.42 x 10^23 kg, and the radius of Mars is 3.40 x 10^6 meters. But we want the distance from the satellite to the surface, so we need to add the radius of Mars to the altitude of the satellite:

r = 3.40 x 10^6 + 175 x 10^3 = 3.575 x 10^6 meters

Now we can calculate the speed:

v = √[(6.67430 × 10^-11) x (6.42 x 10^23) / (3.575 x 10^6)]

Calculating this monstrous equation gives us the speed of the satellite. Hold on to your nose, here we go:

v ≈ 2.9825 x 10^3 meters per second

Now, let's move on to find the period of the satellite. The period represents the time it takes for the satellite to complete one orbit around Mars. We can find it using the following formula:

T = 2π(r/v)

Where:
T = period (in seconds)
r = distance between the center of the planet and the satellite (in meters)
v = speed of the satellite (in meters per second)

Let's plug in the values:

T = 2π x (3.575 x 10^6) / (2.9825 x 10^3)

Simplify, simplify, simplify! And here's your answer:

T ≈ 3.781 x 10^3 seconds

So, the speed of the satellite is approximately 2.9825 x 10^3 meters per second, and the period is roughly 3.781 x 10^3 seconds. I hope I didn't make you space out during this calculation!

To find the speed and period of a satellite orbiting Mars, we can use the following formulas:

1. Centripetal Force (Fc) = Gravitational Force (Fg)
2. Centripetal Force (Fc) = (mass of the satellite) * (acceleration due to gravity)
3. Centripetal Force (Fc) = (mass of the satellite) * (velocity^2) / (radius of the orbit)

We can set up and solve these equations step by step.

Step 1: Calculate the acceleration due to gravity on Mars.
From the formula Fg = G * (mass of Mars) * (mass of the satellite) / (radius^2), where G is the gravitational constant,
we can rearrange the formula to find the acceleration due to gravity (g).
g = (G * mass of Mars) / (radius^2)
= (6.67430 x 10^-11 N*m^2/kg^2 * 6.42 x 10^23 kg) / (3.40 x 10^6 + 1.75 x 10^5)^2
≈ 3.72 m/s^2

Step 2: Calculate the speed of the satellite.
Using the formula Fc = (mass of the satellite) * (velocity^2) / (radius of the orbit), we can rearrange it to solve for velocity (v).
Fc = (mass of the satellite) * (velocity^2) / (radius of the orbit)
G * (mass of Mars) * (mass of the satellite) / (radius^2) = (mass of the satellite) * (velocity^2) / (radius of the orbit)
G * (mass of Mars) * (1) / (radius of the orbit) = (velocity^2)
velocity = √(G * (mass of Mars) / (radius of the orbit))
velocity = √(6.67430 x 10^-11 N*m^2/kg^2 * 6.42 x 10^23 kg) / (3.40 x 10^6 + 1.75 x 10^5)
velocity ≈ 3.25 km/s

Step 3: Calculate the period of the satellite's orbit.
The period (T) is the time it takes for the satellite to complete one orbit around Mars.
T = (2 * π * r) / v, where r is the radius of the orbit and v is the velocity.
T = (2 * π * (3.40 x 10^6 + 1.75 x 10^5) m) / (3.25 x 10^3 m/s)
T ≈ 6548.55 seconds or approximately 1.82 hours

Therefore, the speed of the satellite is approximately 3.25 km/s, and the period of its orbit is approximately 1.82 hours.

To find the speed and period of a satellite orbiting Mars, we can use the formulas related to the gravitational force and centripetal force.

Step 1: Find the gravitational force between Mars and the satellite.
The formula for gravitational force is given by:
F = (G * m1 * m2) / r^2
Where:
- F is the gravitational force between Mars and the satellite.
- G is the gravitational constant (6.67430 x 10^-11 m^3 / (kg * s^2)).
- m1 and m2 are the masses of Mars and the satellite, respectively.
- r is the distance between the center of Mars and the satellite.

Here, the mass of Mars (m1) is given as 6.42 x 10^23 kg. The mass of the satellite (m2) is not provided, so we assume it is much smaller than the mass of Mars and can be ignored.

Given that the satellite orbits 175 km (or 175,000 meters) above the surface of Mars, we add this distance to the average radius of Mars (3.40 x 10^6 meters) to find the total distance (r) between the center of Mars and the satellite.

r = 3.40 x 10^6 + 175,000 = 3.575 x 10^6 meters

Now we can calculate the gravitational force (F) between Mars and the satellite using the formula above.

Step 2: Find the centripetal force acting on the satellite.
The centripetal force is given by:
F = (m * v^2) / r
Where:
- F is the centripetal force acting on the satellite.
- m is the mass of the satellite (again assuming it is much smaller than Mars and can be ignored).
- v is the velocity (speed) of the satellite.
- r is the distance between the center of Mars and the satellite.

We can now equate the gravitational force (F) from step 1 to the centripetal force (F) from step 2 and solve for the velocity (v) of the satellite.

(G * m1 * m2) / r^2 = (m * v^2) / r

Since the mass of the satellite (m) is in both numerator and denominator, we can cancel it out.

(G * m1 * m2) / r = v^2

Simplifying the equation further, we get:

v = sqrt((G * m1) / r)

Now we have the formula to calculate the velocity (speed) of the satellite.

Step 3: Calculate the speed and period of the satellite.
To find the speed of the satellite, substitute the values of G, m1, and r into the equation.

G = 6.67430 x 10^-11 m^3 / (kg * s^2)
m1 = 6.42 x 10^23 kg
r = 3.575 x 10^6 meters

v = sqrt((6.67430 x 10^-11 * 6.42 x 10^23) / (3.575 x 10^6))

After performing the calculation, we will have the speed of the satellite.

To find the period (T) of the satellite, we can use the formula:

T = (2 * π * r) / v

Substitute the values of π, r, and v into the equation.

π = 3.14159
r = 3.575 x 10^6 meters
v = calculated speed of the satellite

After performing the calculation, we will have the period (T) of the satellite.

By following these steps and using the provided formulas and values, you will be able to find the speed and period of a satellite that orbits Mars 175 km above its surface.