The horizontal displacement of a projectile launched at an angle
Answer:
F = GMmx/[√(a² + x²)]³
Explanation:
The force dF on the mass element dm of the ring due to the sphere of mass, m at a distance L from the mass element is
dF = GmdM/L²
Since the ring is symmetrical, the vertical components of this force cancel out leaving the horizontal components to add.
So, the horizontal components add from two symmetrically opposite mass elements dM,
Thus, the horizontal component of the force is
dF' = dFcosФ where Ф is the angle between L and the x axis
dF' = GmdMcosФ/L²
L² = a² + x² where a = radius of ring and x = distance of axis of ring from sphere.
L = √(a² + x²)
cosФ = x/L
dF' = GmdMcosФ/L²
dF' = GmdMx/L³
dF' = GmdMx/[√(a² + x²)]³
Integrating both sides we have
∫dF' = ∫GmdMx/[√(a² + x²)]³
∫dF' = Gm∫dMx/[√(a² + x²)]³ ∫dM = M
F = GmMx/[√(a² + x²)]³
F = GMmx/[√(a² + x²)]³
So, the force due to the sphere of mass m is
F = GMmx/[√(a² + x²)]³
These are the Kepler's laws of planetary motion.
This law relates a planet's orbital period and its average distance to the Sun. - Third law of Kepler.
The orbits of planets are ellipses with the Sun at one focus. - First law of Kepler.
The speed of a planet varies, such that a planet sweeps out an equal area in equal time frames. - Second law of Kepler.
Answer:
The combined gas equation relates three variables pressure, temperature and volume when the number of moles is constant.
The equation is PV / T = constant. Which is valid for a fixed number of moles of the gas.
You can derive the combined gas equation from the combination of Bolye's law, Charles' law and Gay-Lussac's law, which needs some algebra.
Explanation:
The kinetic energy is 
.
Explanation:
The kinetic energy of an object is given by
where
K is the kinetic energy of the object
m is the mass of the object
v is the speed of the object
For the comet in this problem, we have:
is its mass
is the speed
First, we convert the speed from km/h to m/s:
Therefore, the kinetic energy of the comet is
Learn more about kinetic energy here:
brainly.com/question/6536722
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