Besides analyzing collisions, I've seen a derivation of the relativistic mass that goes something like this:
Take gravity to be a constant, downward acceleration. There is a long, straight pole parallel to the ground (and perpendicular to the gravitational acceleration) which is held up by...
Inside the Earth (assuming it is a sphere of uniform density):
g = g_0 \frac{r}{R}
where:
g_0 = -GM / R^2 which is the acceleration due to gravity at the surface of the Earth (M is Earth's mass)
r is the distance from the center of the earth
R is the radius of the earth.
Look at the Lorentz Transformation for the time separating two events:
\Delta t'=\gamma (\Delta t-v\Delta x/c^2)
The primed frame sees the events reversed in time (i.e. \Delta t' is negative) when v\Delta x/c^2 >\Delta t.
Restated:
\Delta t'<0 when v\Delta x/c^2 >\Delta t.
Rearranging the...
Find the relativistic kinetic energy by finding the work done on a particle:
\Delta E_k = W = \int\mathbf{F}\cdot d\mathbf{s} = \int \frac{d\boldsymbol{p}}{dt}\cdot d\mathbf{s}=\int \mathbf{v}\cdot d\mathbf{p} = \int \mathbf{v}\cdot d(\frac{m\mathbf{v}}{\sqrt{1-v^2/c^2}})
After you solve...
Because the mass that is pulling on you is a function of how far you are from the center. If you are at a distance r from the planet's center, the only mass that effects you gravitationally is the mass contained by a (imaginary) spherical surface of radius r. The rest of the mass of the planet...
Thank you for the read. An edited version of my earlier statement should be: "There is no evidence at this time to suggest that antimatter responds to gravity any differently than matter."
This is true for particles that annihilate at rest. Conservation of energy demands that any kinetic energy that the particles had before they annihilated must remain, i.e. if a positron and an electron came together with 1 unit of kinetic energy each, then the two resultant photons after...
That gives you g at any point INSIDE of the planet, assuming constant density. Remember that when you're outside of a spherical object (of uniform density) the mass of the entire thing can be treated as if it were all concentrated at a single point at its center, so once you're outside of the...