The Weight of an object varies inversely as the square of its distance from the centre of the earth.the weight of a spacecraft is observed to be only a quater of its weight on earth.How far is the spacecraft from the surface of the earth?

[take radius of the earth to be 6400km ]

since the weight varies as 1/d^2, if its weight is 1/4 as much, d is twice as big.

That is, 1/(2d)^2 = 1/4 * 1/d^2

To solve this problem, we can use the inverse square law formula:

Weight = k / Distance^2

where k is a constant.

Given that the weight of the spacecraft is observed to be only a quarter of its weight on Earth, we can write:

Weight on Earth = k / Distance^2
Weight on Spacecraft = (1/4) * Weight on Earth = (1/4) * (k / Distance^2)

Let's label the distance between the spacecraft and the center of the Earth as d.

We know that the radius of the Earth, r, is 6400 km. Therefore, the distance between the spacecraft and the surface of the Earth is (r + d).

Now, we can set up the equation and solve for d:

(1/4) * (k / (r + d)^2) = k / d^2

Cross multiplying:

d^2 = (1/4) * (r + d)^2

Expanding and simplifying:

d^2 = (1/4) * (r^2 + 2rd + d^2)

Multiplying both sides by 4:

4 * d^2 = r^2 + 2rd + d^2

3 * d^2 = r^2 + 2rd

Rearranging:

3 * d^2 - 2rd - r^2 = 0

We can solve this quadratic equation by using the quadratic formula:

d = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 3, b = -2r, and c = -r^2.

Substituting the values:

d = (-(-2r) ± √((-2r)^2 - 4 * 3 * (-r^2))) / (2 * 3)

Simplifying:

d = (2r ± √(4r^2 + 12r^2)) / 6

d = (2r ± √16r^2) / 6

d = (2r ± 4r) / 6

We can simplify further:

d = (6r) / 6 = r

Therefore, the spacecraft is located at a distance from the surface of the Earth equal to the radius of the Earth, which is 6400 km.

To solve this problem, we can use the inverse square law relationship between weight and distance from the center of the earth. The formula is given as:

Weight = k / (distance^2)

Where:
- Weight is the weight of the spacecraft
- k is the proportionality constant (we don't need its specific value in this problem)
- distance is the distance from the center of the earth

We know that the weight of the spacecraft is observed to be only a quarter (1/4) of its weight on earth. Let's denote the weight on earth as Wearth. Therefore, we have:

Weight = (1/4) * Wearth

Plugging this into our formula, we get:

(1/4) * Wearth = k / (distance^2)

Now we can rearrange this equation to solve for the distance:

distance^2 = k / ((1/4) * Wearth)

Simplifying:

distance^2 = 4k / Wearth

Taking the square root of both sides, we get:

distance = √(4k / Wearth)

Keep in mind that k is the proportionality constant and we don't have its specific value. However, given the information in the question, we can find the answer without knowing its value. Let's substitute the known values into the equation:

distance = √(4k / Wearth)

Substituting the known values:
- Wearth is the weight on earth, which is given as the weight of the spacecraft
- The radius of the earth is given as 6400 km, so the distance from the spacecraft to the surface of the earth is equal to the radius of the earth plus the unknown distance from the surface of the earth to the spacecraft.

distance = √(4k / Weight) [1]

Since the radius of the earth is 6400 km, the distance from the spacecraft to the center of the earth is equal to the radius plus the unknown distance:

distance = 6400 + d

Now, we can substitute this value into equation [1]:

6400 + d = √(4k / Weight)

Squaring both sides:

(6400 + d)^2 = 4k / Weight

Expanding the left side:

40960000 + 12800d + d^2 = 4k / Weight

Simplifying, we have:

d^2 + 12800d + (40960000 - (4k / Weight)) = 0

Now we have a quadratic equation in terms of the unknown distance d. We can solve this equation using the quadratic formula:

d = (-b ± √(b^2 - 4ac)) / (2a)

For our quadratic equation, the coefficients are:
a = 1
b = 12800
c = 40960000 - (4k / Weight)

Substituting these values into the quadratic formula, you can find the two possible values for d, which will give you the distances from the surface of the earth to the spacecraft.