Compute the altitude above the earth's surface at which the acceleration due to gravity would be 4.9 ms-2? [Assume the mean radius of the earth is 6.4×106 metres and the acceleration due to gravity 9.8 ms-2 on the surface of the earth
The acceleration due to gravity at any point above the earth's surface can be approximated using the formula:
acceleration = (gravity at surface) * (radius / (radius + altitude))^2
Given that the acceleration due to gravity at the surface is 9.8 m/s^2 and the radius of the earth is 6.4 × 10^6 meters, we can rearrange the formula to solve for the altitude:
altitude = (radius + altitude) * sqrt((gravity at surface) / acceleration)
Plugging in the values:
altitude = (6.4 × 10^6 + altitude) * sqrt(9.8 / 4.9)
altitude = (6.4 × 10^6 + altitude) * sqrt(2)
altitude^2 = (6.4 × 10^6 + altitude)^2 * 2
Simplifying further using the binomial expansion:
altitude^2 = (6.4 × 10^6)^2 * 2 + 2 * (6.4 × 10^6) * altitude + altitude^2 * 2
altitude^2 - 2 * (6.4 × 10^6) * altitude - (6.4 × 10^6)^2 * 2 = 0
This equation is a quadratic equation. We can solve it using the quadratic formula:
altitude = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -2 * (6.4 × 10^6), and c = -(6.4 × 10^6)^2 * 2.
Plugging in the values:
altitude = (2 * (6.4 × 10^6) ± sqrt((2 * (6.4 × 10^6))^2 - 4 * 1 * (-(6.4 × 10^6)^2 * 2))) / (2 * 1)
Simplifying further:
altitude = (12.8 × 10^6 ± sqrt((12.8 × 10^6)^2 + 4 * 2 * (6.4 × 10^6)^2)) / 2
altitude = (12.8 × 10^6 ± sqrt((12.8)^2 * (10^6)^2 + 4 * 2 * (6.4)^2 * (10^6)^2)) / 2
altitude = (12.8 × 10^6 ± sqrt((12.8)^2 * (10^6)^2 + 4 * 2 * (6.4)^2 * (10^6)^2)) / 2
altitude = (12.8 × 10^6 ± sqrt((12.8)^2 * (10^6)^2 + 4 * 2 * (6.4)^2 * (10^6)^2)) / 2
altitude = (12.8 × 10^6 ± sqrt((12.8)^2 + 4 * 2 * (6.4)^2) * (10^6)^2) / 2
altitude = (12.8 × 10^6 ± sqrt(163.84 + 4 * 2 * 40.96) * (10^6)^2) / 2
altitude = (12.8 × 10^6 ± sqrt(163.84 + 327.68) * (10^6)^2) / 2
altitude = (12.8 × 10^6 ± sqrt(491.52) * (10^6)^2) / 2
altitude = (12.8 × 10^6 ± 22.15 × 10^6) / 2
Therefore, the two possible altitudes are:
altitude1 = (12.8 × 10^6 + 22.15 × 10^6) / 2 = 35.95 × 10^6 meters
altitude2 = (12.8 × 10^6 - 22.15 × 10^6) / 2 = -4.175 × 10^6 meters
Since a negative altitude does not make sense in this context, we can conclude that the altitude above the earth's surface at which the acceleration due to gravity would be 4.9 m/s^2 is approximately 35.95 × 10^6 meters.