The acceleration due to gravity is inversely proportional to the square of the distance from the centre of Earth. The acceleration due to gravity for a satellite orbiting 7000 km above the centre of Earth is 8.2 m/s^2.
a) write a formula for this relationship.
b) at what height will the acceleration due to gravity be 6 m/s^2?
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I do not know how to approach this question because.. well all I know is that this function needs to be a rational function because this is the topic we are on. Please help and describe why you are doing the steps. Thanks in advance!!
a) The formula for the relationship between acceleration due to gravity (g) and the distance from the center of the Earth (D) is:
g = k / D^2
Where k represents the constant of proportionality.
b) To find the height at which the acceleration due to gravity is 6 m/s^2, we can plug in the given values into the formula and solve for D.
g = 6 m/s^2
D = unknown
Using the formula, we have:
6 = k / D^2
As we don't know the value of the constant of proportionality (k), we can't directly solve for D. However, we can use another equation to relate the acceleration due to gravity at different heights.
Given that the acceleration due to gravity for a satellite orbiting 7000 km above the center of the Earth is 8.2 m/s^2, we can plug these values into the formula:
8.2 = k / (7000 km)^2
Now we have two equations:
6 = k / D^2
8.2 = k / (7000 km)^2
Dividing the second equation by the first equation, we get:
8.2 / 6 = (k / (7000 km)^2) / (k / D^2)
Simplifying, we have:
8.2 / 6 = D^2 / (7000 km)^2
To find D, we can solve for it algebraically:
D^2 = (8.2 / 6) * (7000 km)^2
Taking the square root of both sides:
D = sqrt((8.2 / 6) * (7000 km)^2)
D ≈ 7996 km
Therefore, at a height of approximately 7996 km, the acceleration due to gravity will be 6 m/s^2.
To solve this problem, we can use the inverse square relationship between the acceleration due to gravity and the distance from the center of the Earth. We can express this relationship using the following formula:
g = k / r^2
where g is the acceleration due to gravity, r is the distance from the center of the Earth, and k is a constant of proportionality.
To determine the value of k, we can substitute the given values into the formula. Given that g = 8.2 m/s^2 when r = 7000 km (or 7000000 m), we can write the equation as:
8.2 = k / (7000000)^2
To find k, we can rearrange the equation:
k = 8.2 * (7000000)^2
Simplifying this expression gives us the value of k.
a) The formula for the relationship between the acceleration due to gravity and the distance from the center of the Earth is:
g = (8.2 * (7000000)^2) / r^2
b) To find the height at which the acceleration due to gravity is 6 m/s^2, we can use the formula above and solve for r. We have the value of g (6 m/s^2), and we need to find the corresponding value of r.
6 = (8.2 * (7000000)^2) / r^2
To solve for r, we can rearrange the equation:
r^2 = (8.2 * (7000000)^2) / 6
r^2 = (8.2 * 49000000000000) / 6
Now, we can take the square root of both sides to find r:
r = sqrt((8.2 * 49000000000000) / 6)
Calculating this expression will give you the value of r, which represents the height at which the acceleration due to gravity is 6 m/s^2.
To solve this problem, we can use the concept of inverse variation. Inverse variation states that when two variables are inversely proportional, their product is a constant.
a) To write a formula for the relationship between the acceleration due to gravity (g) and the distance from the center of the Earth (r), we can use the inverse variation equation:
g × r^2 = k
Where g is the acceleration due to gravity, r is the distance from the center of the Earth, and k is the constant of variation.
b) To find the height at which the acceleration due to gravity is 6 m/s^2, we can substitute the given values into the equation:
6 × r^2 = k
To solve for r, we need to find the value of k. We can use the information given for the satellite orbiting 7000 km above the center of the Earth, where the acceleration due to gravity is 8.2 m/s^2. Plugging in these values, we have:
8.2 × 7000^2 = k
Solving for k, we get k ≈ 404,600,000.
Now we can substitute this value of k into the equation for part (b):
6 × r^2 = 404,600,000
Divide both sides by 6 to isolate r^2:
r^2 = 67,433,333.33
Finally, take the square root of both sides to find r:
r ≈ √67,433,333.33
Therefore, the height at which the acceleration due to gravity is 6 m/s^2 is approximately 8,207.5 km above the center of the Earth.
g(d)=K/d^2
where
d= distance in km,
g= acceleration in m/s^2
Use the following relationship to find K
g(7000)=8.2 m/s^2
then use the calculated value of K to find
g(x)=6 m/s^2
addendum:
then use the calculated value of K to find x, where x is the distance from the centre of the earth to be found.
g(x)=6 m/s^2