in 2005, the space probe Deep Impact launched a 30 kg projectile into Comet Tempel 1. Observing the collision helped scientists learn about the comet’s characteristics. The come is estimated to have a mass of 9.0 x 1013 kg.
a. Assuming the estimated mass of the comet at that time was correct, at what distance from the comet’s centre was the gravitational force between the comet and the projectile 32 N?
b. What was the magnitude of the gravitational force between the comet and the projectile at a distance of 350 m?
c. Deep Impact also released a probe to fly by the comet and record images of the collision. Determine the strength of the comet’s gravitational field at the probe’s distance of 5.0 x 103 km from the comet.
F=G•m•M/R².
(a) R =sqrt (G•m•M/F)
(b) F1=G•m•M/(R1)²
(c) g= G•M/(R2)²
a. To find the distance from the comet's center where the gravitational force between the comet and the projectile is 32 N, we can use the formula for gravitational force:
F = G * (m1 * m2) / r^2
where F is the force, G is the gravitational constant (approximately 6.67430 × 10^-11 N m^2 / kg^2), m1 and m2 are the masses of the objects (comet and projectile in this case), and r is the distance between their centers.
In this case, the mass of the projectile (m1) is 30 kg and the mass of the comet (m2) is 9.0 x 10^13 kg. The force (F) is 32 N.
The formula can be rearranged to solve for r:
r = sqrt((G * m1 * m2) / F)
Substituting the values:
r = sqrt((6.67430 × 10^-11 N m^2 / kg^2 * 30 kg * 9.0 x 10^13 kg) / 32 N)
Simplifying and calculating:
r ≈ 6.54 × 10^9 meters
Therefore, the distance from the comet's center where the gravitational force between the comet and the projectile is 32 N is approximately 6.54 × 10^9 meters.
b. To find the magnitude of the gravitational force between the comet and the projectile at a distance of 350 m, we can use the same formula as above:
F = G * (m1 * m2) / r^2
Substituting the values:
F = (6.67430 × 10^-11 N m^2 / kg^2 * 30 kg * 9.0 x 10^13 kg) / (350 m)^2
Calculating:
F ≈ 2.44 × 10^12 N
Therefore, the magnitude of the gravitational force between the comet and the projectile at a distance of 350 m is approximately 2.44 × 10^12 N.
c. To determine the strength of the comet's gravitational field at a distance of 5.0 x 10^3 km from the comet, we need to calculate the gravitational field strength using the formula:
g = G * M / r^2
where g is the gravitational field strength, G is the gravitational constant, M is the mass of the object (comet), and r is the distance from the center of the object.
In this case, we have the mass of the comet (M) as 9.0 x 10^13 kg and the distance (r) as 5.0 x 10^3 km (which needs to be converted to meters).
Converting km to meters:
r = 5.0 x 10^3 km * 10^3 m/km
Substituting the values into the formula:
g = (6.67430 × 10^-11 N m^2 / kg^2 * 9.0 x 10^13 kg) / (5.0 x 10^3 km * 10^3 m/km)^2
Calculating:
g ≈ 2.68 × 10^-2 N/kg
Therefore, the strength of the comet's gravitational field at a distance of 5.0 x 10^3 km from the comet is approximately 2.68 × 10^-2 N/kg.