Kepler's second law states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This law is a direct consequence of which conservation principle?
Physics
Question · Easy
Factor it
Short
A hypothetical planet has twice the mass of Earth and twice the radius of Earth. If the escape velocity from Earth's surface is $11.2\text{ km/s}$, what is the escape velocity from the surface of this hypothetical planet?
Question · Medium
A satellite is in a circular orbit around Earth at an altitude equal to Earth's radius $R_E$. If the orbital speed of a satellite near the Earth's surface is $v_0 \approx 7.9\text{ km/s}$, what is the orbital speed of this satellite?
What is the time period of a simple pendulum of length $L$ if it is placed inside a spacecraft orbiting the Earth in a stable circular path of radius $2R_E$?
A cosmic particle of mass $m$ undergoes a head-on elastic collision with a stationary helium nucleus of mass $4m$. What fraction of the particle's initial kinetic energy is transferred to the helium nucleus?
Star A has radius $R$ and surface temperature $2T$. Star B has radius $2R$ and surface temperature $T$. Assuming both behave as perfect blackbodies, what is the ratio of the luminosity of Star A to that of Star B ($L_A / L_B$)?
A star has a surface temperature of $6000\text{ K}$. Assuming it radiates as a blackbody, at what wavelength does its spectral radiance peak? (Wien's displacement constant $b \approx 2.898 \times 10^{-3}\text{ m K}$)
A disk-shaped galaxy of mass $M$ and radius $R$ is rotating with constant angular velocity $\omega$. If it collapses uniformly under internal gravitational forces to half its original radius while preserving its disk shape and mass, what is its new angular velocity?
Question · Hard
An astronomical telescope has an objective lens of focal length $100\text{ cm}$ and an eyepiece of focal length $2.0\text{ cm}$. Under normal adjustment (focused at infinity), what is the magnification and the total length of the telescope tube?
What is the theoretical angular resolution (in radians) of the Hubble Space Telescope for visible light of wavelength $\lambda = 500\text{ nm}$, given its primary mirror diameter is $D = 2.4\text{ m}$?
Porro prisms are used in binoculars to invert the image without changing its size. What physical phenomenon enables these prisms to reflect light with 100% efficiency?
In a Young's double-slit experiment, the slit separation is $d = 0.5\text{ mm}$ and the distance to the screen is $D = 1.0\text{ m}$. If light of wavelength $\lambda = 600\text{ nm}$ is used, what is the width of the interference fringes?
The work function of a certain metal surface is $2.0\text{ eV}$. What is the threshold frequency of light required to eject photoelectrons from this surface? (Planck's constant $h \approx 6.63 \times 10^{-34}\text{ J s}$, $1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$)
What is the wavelength of the photon emitted when an electron in a hydrogen atom transitions from the $n=3$ state to the $n=2$ state (first line of the Balmer series)? (Rydberg constant $R_H \approx 1.097 \times 10^7\text{ m}^{-1}$)
A galaxy is observed to have its Hydrogen-alpha spectral line ($656.3\text{ nm}$) redshifted to $721.9\text{ nm}$. What is the radial velocity of the galaxy relative to Earth?
A proton of velocity $v = 3.0 \times 10^6\text{ m/s}$ enters Earth's magnetic field ($B = 5.0 \times 10^{-5}\text{ T}$) at an angle of $90^\circ$ to the field lines. What is the magnitude of the magnetic force acting on the proton? ($q = 1.6 \times 10^{-19}\text{ C}$)
Two point charges, $+q$ and $-q$, are separated by a distance $r$ in a vacuum. If the distance between them is doubled, what happens to the electrostatic force between them?
A loop of wire enclosing an area of $0.1\text{ m}^2$ is placed in a uniform magnetic field that decreases from $2.0\text{ T}$ to $0.0\text{ T}$ in $0.5\text{ seconds}$. If the magnetic field is perpendicular to the plane of the loop, what is the induced electromotive force (EMF)?
A planet moves in a circular orbit of radius $r$ around a star of mass $M$ with constant speed $v$. What is the centripetal acceleration of the planet?
The tidal force exerted by a planet of mass $M$ and radius $R$ on a nearby small moon of mass $m$ at distance $r$ scales with distance as which of the following?
A Carnot engine absorbs heat from a hot stellar core reservoir at $10000\text{ K}$ and rejects heat to the surrounding stellar envelope at $3000\text{ K}$. What is the thermodynamic efficiency of this engine?
The temperature of the solar corona is about $1.0 \times 10^6\text{ K}$. What is the root-mean-square (RMS) speed of a proton (hydrogen nucleus) at this temperature? ($k_B = 1.38 \times 10^{-23}\text{ J/K}$, proton mass $m_p \approx 1.67 \times 10^{-27}\text{ kg}$)
A white dwarf star collapses uniformly to a neutron star, shrinking its radius by a factor of 100 while maintaining a uniform spherical mass distribution. Assuming no mass is lost, by what factor does its rotational kinetic energy increase if angular momentum is conserved?
Which of Maxwell's equations directly implies the absence of magnetic monopoles in nature?
Two thin lenses of focal lengths $f_1 = +20\text{ cm}$ and $f_2 = -40\text{ cm}$ are placed in contact. What is the equivalent focal length and the nature of the lens combination?
The speed of light in a vacuum is $c \approx 3.00 \times 10^8\text{ m/s}$. If the refractive index of a certain glass prism is $1.50$, what is the speed of light in this glass?
A radioactive isotope has a half-life of 10 days. If a sample initially contains $1.6 \times 10^{20}$ active nuclei, how many active nuclei will remain after 30 days?
A comet moves in a highly elliptical orbit around the Sun. If the ratio of its maximum distance (aphelion) to minimum distance (perihelion) from the Sun is 9, what is the ratio of its speed at perihelion to its speed at aphelion ($v_p / v_a$)?
What is the work required to launch a payload of mass $m$ from the surface of Earth (radius $R_E$) to an altitude equal to Earth's radius $R_E$?
A dust particle of mass $m$ experiencing a drag force $F_d = -kv$ falls through a gas. What is the terminal velocity of the dust particle under gravity $g$?
According to the kinetic theory of gases, the pressure $P$ exerted by an ideal gas of density $\rho$ is related to the root-mean-square velocity $v_{rms}$ by which formula?
Light travelling in air ($n_1 = 1.00$) is reflected from the surface of a flat glass plate of refractive index $n_2 = 1.73$. At what angle of incidence will the reflected light be completely polarized? (Recall $\tan(60^\circ) \approx 1.73$)
The mass of a helium nucleus ($^4_2\text{He}$) is $4.0015\text{ u}$, while the masses of a free proton and neutron are $1.0073\text{ u}$ and $1.0087\text{ u}$ respectively. What is the binding energy of the helium nucleus? ($1\text{ u} \approx 931.5\text{ MeV}$)
In a conductor carrying a steady current, what is the order of magnitude of the average drift velocity of the conduction electrons?
A parallel plate capacitor has capacitance $C_0$ in a vacuum. If a dielectric slab of dielectric constant $K = 5$ is inserted to fill the entire space between the plates, what is the new capacitance?
A projectile is launched from the surface of Mars (gravity $g_M \approx 3.7\text{ m/s}^2$) at an angle of $45^\circ$ to the horizontal with an initial velocity of $10\text{ m/s}$. What is the horizontal range of this projectile, neglecting atmospheric resistance?
In a binary star system, Star 1 has mass $M_1 = 2 M_\odot$ and Star 2 has mass $M_2 = 6 M_\odot$. If the separation between the centers of the two stars is $d$, how far is the center of mass of the system from the center of Star 1?
What are the SI base units of the universal gravitational constant $G$?
Chromatic aberration in lenses is caused by which optical phenomenon?
Which of the following processes is responsible for the transport of energy from the Sun's core to its outer convective zone?
If $\sin\theta + \cos\theta = \sqrt{2}$, what is the value of $\sin(2\theta)$?
Mathematics
What is the value of $\cos(15^\circ) - \sin(15^\circ)$?
An ellipse representing a planetary orbit is described by the equation $9x^2 + 25y^2 = 225$. What is the eccentricity of this orbit?
A parabolic path of a comet is modeled by $y^2 = 4ax$. If a straight line representing a light beam $y = mx + c$ is tangent to the comet's path, what is the relationship between the parameters?
What is the shortest distance from the origin $(0,0,0)$ to the plane $2x - 3y + 6z = 21$?
The position of a particle moving along a straight line is given by $s(t) = 2t^3 - 9t^2 + 12t$. What is the acceleration of the particle at the instant when its velocity is zero?
A rocket's height is described by $h(t) = 40t - 5t^2$ meters, where $t$ is in seconds. What is the maximum height reached by the rocket?
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