A man weighs 900N on the surface of the earth. Compare his weight in an airplane flying at an elevation of 9000m. Note that the radius of Earth is 6378km.
Using gravitational law, F = G(m1)(m2)/r^2, where m1 is the mass of the earth = 5.9742x10^24kg, m2= mass of the man = 900/9.81 = 91.74kg, the r is the distance between the man and the center of the air and it is equal to (6378km+9000m = 6387km), this computation is almost equal to the answer when the solution you use is ratio and proportion.
Well, it sounds like this man is taking his weight to new heights... literally! But let's do some calculations to figure out how his weight changes inside the airplane.
First, we need to consider that weight is determined by the force of gravity acting on an object. The formula for weight is W = mg, where W represents weight, m represents mass, and g represents the acceleration due to gravity.
On the surface of the Earth, the gravitational acceleration is approximately 9.8 m/s². So with a weight of 900N, we can find the mass of the man using the formula:
m = W / g
m = 900N / 9.8 m/s²
m ≈ 91.8 kg
Now, let's calculate his weight in the airplane, which is flying at an elevation of 9000m. We need to consider that as we move away from the Earth's surface, the acceleration due to gravity decreases. This is because the gravitational force weakens with distance. The formula for gravitational acceleration is:
g' = g * (r / (r + h))^2
where g' signifies the new gravitational acceleration, g is the acceleration due to gravity at the Earth's surface, r is the radius of the Earth, and h is the height or elevation above the surface.
Plugging in the values:
g' = 9.8 m/s² * (6378000 m / (6378000 m + 9000 m))^2
g' ≈ 9.66 m/s²
Now, we can calculate the man's weight in the airplane using the same formula as before:
W' = m * g'
W' = 91.8 kg * 9.66 m/s²
W' ≈ 887.9 N
So, in the airplane flying at an elevation of 9000m, the man's weight would be approximately 887.9N. It seems like his weight gets a bit of a reduction during the flight, but he'll still feel grounded... just a little lighter!
To compare the man's weight on the surface of the Earth with his weight in an airplane flying at an elevation of 9000m, we need to consider the concept of gravitational force and how it changes with distance.
Gravitational force is directly proportional to the mass of an object and inversely proportional to the square of the distance between the object and the center of the Earth. So, we can use the formula for gravitational force:
F = (G * m₁ * m₂) / r²
Where:
F is the gravitational force,
G is the gravitational constant (approximately 6.67430 × 10^-11 N*m²/kg²),
m₁ and m₂ are the masses of the two objects (in this case, the man and the Earth),
and r is the distance between the center of the Earth and the man.
First, let's calculate the man's weight on the surface of the Earth using his mass (we assume it's given). Weight is the force of gravity acting on an object:
Weight₁ = m₁ * g
Where:
Weight₁ is the man's weight on the surface of the Earth,
m₁ is the man's mass in kg,
and g is the acceleration due to gravity on Earth's surface (approximately 9.8 m/s²).
Now, let's calculate the man's weight in the airplane flying at an elevation of 9000m. In this case, the distance between the center of the Earth and the man is the sum of the Earth's radius and the elevation:
r₂ = r + h
Where:
r₂ is the new distance between the center of the Earth and the man,
r is the Earth's radius (6378 km or 6378000 m),
and h is the elevation of the airplane (9000 m).
Finally, we can calculate the man's weight in the airplane using the new distance:
Weight₂ = (G * m₁ * m₂) / r₂²
Comparing the two weights will give us the answer to the question.