Tag Archives: Math

Massive episode about relativity

21 Mar

And finally, the conclusion of our four part primer on relativity. Here’s a recap on the topics we’ve covered so far; the laws of physics depend on the frame of reference; the faster an object moves, the slower time moves in that frame; the faster an object moves, the shorter that object appears from a resting frame.  In this episode, we’re look at questions about one more property of an object. Specifically, how does motion affect mass?

Well, if there is one equation that people remember about special relativity, it’s

52c7687643df1c12231b39e324850586

And we’ve established that the speed of light, c, is a constant, so mass just depends on the amount of energy. Shortest fun fact ever…

Unfortunately not. E = mc² is a special solution of the more general equation

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where m should be the mο and p is the object’s momentum (inertial mass [m] x velocity [v]). When the velocity of the object is zero, the momentum expression disappears and we are left with the now famous equation; an equation which describes how the energy stored in an object is related to its resting mass. Hmm. What the heck is inertial mass, and how is it different from rest mass?

Inertia refers to the tendency for an object to continue traveling and resist a force that attempts to change it’s motion.  The rest mass is a measurement of the energy stored in the mass in a resting frame.  The relationship between the rest mass and the inertial mass (sometimes called the effective or relative mass) is

images

where mο is the rest mass and m is the effective mass.  As a result the effective/inertial mass of an object mass increases as the object travels at increasing speed.

Now, let’s put it all together.

  • An object can’t travel fast than the speed of light.
  • As an object’s velocity get’s closer to the speed of light, it’s effective mass increases.
  • In order to conserve resting mass in the Energy-Momentum relation,  energy and momentum vary proportionally.

With these conditions, adding energy to an object and attempting to increase it’s velocity results in an increase in it’s effective mass that grows exponentially as the velocity approaches c.

exponential-growth-graph

This is what exponential growth looks like. Small change initially; infinite change as v approaches c.

One cool implication of this is that if you keep putting energy into an object, it could achieve infinite apparent mass. Before this point it would form a black hole, but it’s still cool to think about.

I think that’s enough mind blowing stuff about relativity. Don’t worry if it seems a little confusing. To adapt a phrase from Niels Bohr, “Anyone who is not shocked by special relativity has not understood it.” If there’s nothing more that you take from these last few episodes, I hope you at least will accept that the most famous equation in physics really ought to be written as Eο = mοc2.

Keep moving, you’ll stay younger

16 Mar

So in the last episode, I explained very basically what relativity means. We saw that the same event could appear differently to two observers depending each man’s frame of reference. That was bizarre, and from there we could begin to ask other questions. For instance, what other measurements depend on our frame of reference?

Let’s start with one of the more basic measurements-time. We’re going to use another example to explain what’s happening, but this time we’re also going to use some algebra and a triangle. Bear with me, the math isn’t too bad.

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

Alright, so imagine that we’ve got a space ship that’s traveling at speed v, and there’s a flash of light that is bouncing up and down inside of the space ship. The speed of  the flash of light (and all light for that matter) is c, and that doesn’t change because the speed of light is a constant. So far so good? Great.

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

For an observer on the ship, we’re at rest with respect to the ship. Another way of saying that is that the ship is his rest frame. If we call the height of the ship L, then the total time it takes the light to travel from the floor to the ceiling is

image004

Source: PhysicsForIdiots.com

Source: PhysicsForIdiots.com

But looking up at the ship from Earth is a different story because we’re dealing with a different frame of reference. While the light is bouncing up and down, it also has to travel horizontally with the rest of the ship. The total distance it travels is the diagonal, D. As a result, the time it takes in this alternate frame is.

image005

Note that the speed of light, c, is the same for both of these equations. D, however, is not the same as L; D is longer than L.

Let’s stop right here to drive home that concept. The light travels less distance in the rest frame than it does in the moving frame. As a result, less time passes in the rest frame than in the moving frame. A clock on the space ship would go slower than a clock on Earth. That’s time dilation, and it’s not just a theory; It’s a fact!

… Now, if you wanted to find out how much slower, you’d have to do some math. We can find D using the Pythagorean theorem

image006

At this point the math get’s really elegant (don’t worry, I won’t show it). Suffice it to say, you plug the D into the Δt’ equation, square the whole expression, collect the Δt’ terms, isolate the Δt’, square root the whole expression, and you end up with this equation

image007

Which is cool, because then we can substitute in Δt to get

image008…the equation for how time in a moving frame relates to time in a rest frame. Boom.

So that’s time travel. Questions are totally appropriate at this time. Stay tuned for the next episode where I’ll show you how to shrink objects.