An asteroid in space is attracted to the planet XYZ such that it accelerates at 0.0085 m/s^2 towards it. Given that planet XYZ has a mass of 4.7*10^24 kg. How far out from the planet is the asteroid?
Force=mass*acceleration
GMm/d^2=m a
d^2=GM/a M=4.7E24, a given, G known.
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To determine the distance from the planet XYZ at which the asteroid is located, we can use the concept of gravitational acceleration and the formula for gravitational force.
The formula to calculate gravitational force is given by:
F = (G * m₁ * m₂) / r²
Where:
F is the gravitational force between two objects,
G is the gravitational constant (approximately 6.67430 × 10⁻¹¹ N m²/kg²),
m₁ and m₂ are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, the gravitational force experienced by the asteroid is equal to the product of the asteroid's mass (m) and the gravitational acceleration (g).
F = m * g
Given that the asteroid's acceleration towards planet XYZ is 0.0085 m/s², we can equate this with the gravitational force to find the mass of the asteroid.
m * g = (G * m * M) / r²
Where:
M is the mass of planet XYZ.
Simplifying the equation, we get:
g = (G * M) / r²
Rearranging the equation to solve for r, we have:
r = √((G * M) / g)
Now, let's substitute the known values into the equation:
G = 6.67430 × 10⁻¹¹ N m²/kg²
M = 4.7 × 10²⁴ kg
g = 0.0085 m/s²
Calculating the distance from the planet XYZ, we have:
r = √((6.67430 × 10⁻¹¹ N m²/kg² * 4.7 × 10²⁴ kg) / 0.0085 m/s²)
r = √((6.67430 × 4.7) / 0.0085) × (10⁻¹¹ × 10²⁴ / (m/s²))
r = √(31.34961 / 0.0085) × (10⁻¹¹ × 10²⁴ / (m/s²))
r = √3685.783529 × (10⁻¹¹ × 10²⁴ / (m/s²))
r ≈ √3685.783529 × (10²² / (m/s²))
r ≈ √(3685.783529 × 10²²) / (m/s)
r ≈ √(368578.3529 × 10²²) / (m/s)
r ≈ √(36857835290) / (m/s)
r ≈ 192039.269 m/s
Therefore, the asteroid is approximately 192,039.269 meters (or 192.039 kilometers) away from planet XYZ.