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.
mv²/R=GmM/R²
v=sqrt(GM/R )=sqrt{6.67•10⁻¹¹•6.42•10²³ /(3.40 •10⁶+175 000)}=3.46•10³ m/s.
T=2πR/v=2•3.14•(3.40 •10⁶+175 000)/ 3.46•10³ =6.5•10³ s=1.8 h
To find the speed and period of a satellite orbiting Mars, we can use the following formulas:
1. Gravitational force between two objects: F = (G * m1 * m2) / r^2
2. Centripetal force: F = (m * v^2) / r
3. Gravitational force is equal to centripetal force: (G * m1 * m2) / r^2 = (m * v^2) / r
First, let's calculate the gravitational force between Mars and the satellite:
G = 6.67 x 10^-11 N(m/kg)^2 (gravitational constant)
m1 = 6.42 x 10^23 kg (mass of Mars)
m2 = mass of the satellite (which is not given)
r = radius of Mars + altitude of the satellite = 3.40 x 10^6 m + 175,000 m = 3.575 x 10^6 m
F = (G * m1 * m2) / r^2
Next, let's calculate the centripetal force of the satellite:
m = mass of the satellite (which is not given but cancels out in this equation)
v = velocity of the satellite
r = radius of Mars + altitude of the satellite = 3.40 x 10^6 m + 175,000 m = 3.575 x 10^6 m
F = (m * v^2) / r
Setting the gravitational force equal to the centripetal force:
(G * m1 * m2) / r^2 = (m * v^2) / r
Simplifying:
(G * m1 * m2) = (m * v^2) * r
Rearranging:
v^2 = (G * m1 * m2) / r
Finally, solve for the speed (v):
v = sqrt((G * m1 * m2) / r)
To find the period (T) of the satellite's orbit, we can use the formula:
T = (2 * π * r) / v
To summarize, the steps are as follows:
1. Calculate the gravitational force between Mars and the satellite using equation 1.
2. Calculate the centripetal force of the satellite using equation 2.
3. Set the gravitational force equal to the centripetal force and solve for the speed using equation 3.
4. Calculate the period of the satellite's orbit using the formula T.
Please note that we cannot find the speed and period without knowing the mass of the satellite.
To find the speed and period of a satellite orbiting Mars, we will use the following formulas:
1. Centripetal Force (F) = Gravitational Force (Fg)
2. Gravitational Force (Fg) = (G * m1 * m2) / r^2
3. Centripetal Force (F) = (m * v^2) / r
4. Gravitational Force (Fg) = (m * (2 * π * r) / T)^2 * r
Where:
- F is the centripetal force (in Newtons)
- G is the gravitational constant (6.67430 × 10^-11 N(m/kg)^2)
- m1 and m2 are the masses of the two objects (in kg)
- r is the distance between the two objects (in meters)
- m is the mass of the satellite (in kg)
- v is the speed of the satellite (in m/s)
- T is the period of the satellite (in seconds)
Step 1: Calculating the Gravitational Force (Fg)
We can calculate the gravitational force between Mars and the satellite using the formula Fg = (G * m1 * m2) / r^2.
Given that the mass of Mars is 6.42 x 10^23 kg, the average radius of Mars is 3.40 x 10^6 m, and the satellite is 175 km above the surface (which we can add to the radius of Mars), we can calculate the force.
Fg = ((6.67430 × 10^-11) * (6.42 x 10^23) * m) / (3.4 x 10^6 + 175 x 10^3)^2
Step 2: Calculating the Centripetal Force (F)
We can calculate the centripetal force using the formula F = (m * v^2) / r.
As we know that the gravitational force and the centripetal force are equal, we can set F = Fg and solve for v.
(m * v^2) / r = ((6.67430 × 10^-11) * (6.42 x 10^23) * m) / (3.4 x 10^6 + 175 x 10^3)^2
Step 3: Simplify the Equation and Solve for v
First, cancel out the mass of the satellite (m) on both sides of the equation.
Then, multiply both sides by (3.4 x 10^6 + 175 x 10^3)^2 to isolate v^2 and take the square root to find v.
v^2 = ((6.67430 × 10^-11) * (6.42 x 10^23)) / (3.4 x 10^6 + 175 x 10^3)^2
v = square root of ((6.67430 × 10^-11) * (6.42 x 10^23)) / (3.4 x 10^6 + 175 x 10^3)
Step 4: Calculating the Period (T)
We can use the formula for the gravitational force to calculate the period T.
((6.67430 × 10^-11) * (6.42 x 10^23) * m) / (r^2) = (m * (2 * π * r) / T)^2 * r
Simplifying the equation, we find:
((6.67430 × 10^-11) * (6.42 x 10^23)) / (r^2) = (4 * π^2 * r) / T^2
Now we can solve for T:
T^2 = (4 * π^2 * r^3) / ((6.67430 × 10^-11) * (6.42 x 10^23))
T = square root of ((4 * π^2 * r^3) / ((6.67430 × 10^-11) * (6.42 x 10^23)))
Once you substitute the values of r and the necessary constants into the formulas, you can calculate the speed (v) and period (T) of the satellite orbiting Mars.