Either light is an exception, or the classical velocity addition formula only works at low velocities. The latter is the case. The correct formula for one-dimensional
relativistic velocity addition is
where
$v$ is the relative velocity between two observers,
$u$ is the velocity of an object relative to one observer, and
$u\prime $ is the velocity relative to the other observer. (For ease of visualization, we often choose to measure
$u$ in our reference frame, while someone moving at
$v$ relative to us measures
$u\prime $ .) Note that the term
$\frac{vu\prime}{{c}^{2}}$ becomes very small at low velocities, and
$u=\frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}$ gives a result very close to classical velocity addition. As before, we see that classical velocity addition is an excellent approximation to the correct relativistic formula for small velocities. No wonder that it seems correct in our experience.
Showing that the speed of light towards an observer is constant (in a vacuum): the speed of light is the speed of light
Suppose a spaceship heading directly towards the Earth at half the speed of light sends a signal to us on a laser-produced beam of light. Given that the light leaves the ship at speed
$c$ as observed from the ship, calculate the speed at which it approaches the Earth.
Strategy
Because the light and the spaceship are moving at relativistic speeds, we cannot use simple velocity addition. Instead, we can determine the speed at which the light approaches the Earth using relativistic velocity addition.
Solution
Identify the knowns.
$\text{v=}0\text{.}\text{500}c$ ;
$u\prime =c$
Identify the unknown.
$u$
Choose the appropriate equation.
$u=\frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}$
Relativistic velocity addition gives the correct result. Light leaves the ship at speed
$c$ and approaches the Earth at speed
$c$ . The speed of light is independent of the relative motion of source and observer, whether the observer is on the ship or Earth-bound.
Velocities cannot add to greater than the speed of light, provided that
$v$ is less than
$c$ and
$u\prime $ does not exceed
$c$ . The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.
Comparing the speed of light towards and away from an observer: relativistic package delivery
Suppose the spaceship in the previous example is approaching the Earth at half the speed of light and shoots a canister at a speed of
$0.750c$ . (a) At what velocity will an Earth-bound observer see the canister if it is shot directly towards the Earth? (b) If it is shot directly away from the Earth? (See
[link] .)
Strategy
Because the canister and the spaceship are moving at relativistic speeds, we must determine the speed of the canister by an Earth-bound observer using relativistic velocity addition instead of simple velocity addition.
Solution for (a)
Identify the knowns.
$\text{v=}0.500c$ ;
$u\prime =0\text{.}\text{750}c$
Identify the unknown.
$u$
Choose the appropriate equation.
$\text{u=}\frac{\mathrm{v+u}\prime}{1+\frac{vu\prime}{{c}^{2}}}$
The minus sign indicates velocity away from the Earth (in the opposite direction from
$v$ ), which means the canister is heading towards the Earth in part (a) and away in part (b), as expected. But relativistic velocities do not add as simply as they do classically. In part (a), the canister does approach the Earth faster, but not at the simple sum of
$1.250c$ . The total velocity is less than you would get classically. And in part (b), the canister moves away from the Earth at a velocity of
$-0.400c$ , which is
faster than the
$\mathrm{-0.250}c$ you would expect classically. The velocities are not even symmetric. In part (a) the canister moves
$0.409c$ faster than the ship relative to the Earth, whereas in part (b) it moves
$0.900c$ slower than the ship.
Questions & Answers
Is there any normative that regulates the use of silver nanoparticles?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?