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
is the relative velocity between two observers,
is the velocity of an object relative to one observer, and
is the velocity relative to the other observer. (For ease of visualization, we often choose to measure
in our reference frame, while someone moving at
relative to us measures
.) Note that the term
becomes very small at low velocities, and
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
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.
;
Identify the unknown.
Choose the appropriate equation.
Plug the knowns into the equation.
Discussion
Relativistic velocity addition gives the correct result. Light leaves the ship at speed
and approaches the Earth at speed
. 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
is less than
and
does not exceed
. 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
. (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.
;
Identify the unknown.
Choose the appropriate equation.
Plug the knowns into the equation.
Solution for (b)
Identify the knowns.
;
Identify the unknown.
Choose the appropriate equation.
Plug the knowns into the equation.
Discussion
The minus sign indicates velocity away from the Earth (in the opposite direction from
), 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
. 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
, which is
faster than the
you would expect classically. The velocities are not even symmetric. In part (a) the canister moves
faster than the ship relative to the Earth, whereas in part (b) it moves
slower than the ship.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
Got questions? Join the online conversation and get instant answers!