Meaning of physical quantities and division

[Pilpellon]

I read this interesting thread trying to find an answer to my questions (and I got even more confused). I study radiometry units in context of computer graphics.
I have a few questions, starting from the basic ones:
1. Following AlexS's note, why is speed=distance/time and not distance*time?
2. Following the definition of speed, as in speed = how much distance passed for some time, I still get confused sometimes when power operators come in, for example acceleration = distance / seconds^2, or even worse when force=kg*m/s^2. Is it possible to straight on understand the logic of the conjunction of all these units?
3. In my book, they say that differential irradiance=radiance*differential solid angle*cos(theta). Say I did get the meaning of division in physics (how much x for y, kinda..), but what does multiplication mean? Aside from the fact in order to get the irradiance unit correct, the radiance must be multiplied by a solid angle, but what does that multiplication mean?
[Mentor's note: One unrelated question moved to its own thread]

I know some of these questions might sound silly to you, but they have been in my head for a long time, so straightforward answers will sure help.

 

[A.T.]

why is speed=distance/time and not distance*time?

More time for same distance is faster or slower?

 

[Jehannum]

I read this interesting thread trying to find an answer to my questions (and I got even more confused). I study radiometry units in context of computer graphics.
I have a few questions, starting from the basic ones:
1. Following AlexS's note, why is speed=distance/time and not distance*time?
2. Following the definition of speed, as in speed = how much distance passed for some time, I still get confused sometimes when power operators come in, for example acceleration = distance / seconds^2, or even worse when force=kg*m/s^2. Is it possible to straight on understand the logic of the conjunction of all these units?
3. In my book, they say that differential irradiance=radiance*differential solid angle*cos(theta). Say I did get the meaning of division in physics (how much x for y, kinda..), but what does multiplication mean? Aside from the fact in order to get the irradiance unit correct, the radiance must be multiplied by a solid angle, but what does that multiplication mean?
[Mentor's note: One unrelated question moved to its own thread]

I know some of these questions might sound silly to you, but they have been in my head for a long time, so straightforward answers will sure help.

1. Calculating speed as distance traveled / time taken allows you to work out useful things like how long the object will take to go a certain distance, or how far it will go in a certain time.

If an object travels 20 metres in 5 seconds, it's (average) speed is 20 / 5 = 4 metres per second.

This is easy to understand intuitively: every second the object will travel 4 metres. Therefore if you know how many seconds, you can calculate how many metres.

You wouldn't be able to do this with distance traveled x time taken. It wouldn't give you a useable quantity. To have a quantity that's useable in equations you need to reduce one of the values to 1 (in this case it's time, i.e. 1 second).

 

[Jehannum]

2. If you understand speed as discussed above, you should be ready to understand acceleration.

You can quantify a change in speed as how much the speed changes per second.

For example, if an object's speed changes from 0 m / s to 27 m / s in 8 seconds, every second the speed increase will be 27 / 8 = 3.375 m / s on average.

This rate of change of speed is called acceleration.

Acceleration is how much m / s changes per second. This can be written as m / s / s.

Simple laws of mathematics (powers) tell us that m / s / s = (m / s) x (1 / s) = m / s^2, so we can use that form as an alternative for writing the unit of acceleration.

Knowing acceleration allows us to calculate what the speed would be after, say, 100 seconds, starting from rest and assuming acceleration is constant:

100 s x 3.375 m / s^2 = 337.5 m / s.

 

[Jehannum]

3. I have no idea of this stuff, so I'll leave it there. :)

 

[jbriggs444]

3. In my book, they say that differential irradiance=radiance*differential solid angle*cos(theta). Say I did get the meaning of division in physics (how much x for y, kinda..), but what does multiplication mean?

Multiplication is easy. If you have ten grapes that mass three grams per grape then you have thirty grams of grapes. 10*3 = 30.

Now back to the irradiance = radiance * differential solid angle * cos(theta)...

If you have an light bulb with a radiance of 3 Watts per steradian hitting your handkerchief and you crumple the handkerchief so that it covers a solid angle that is 0.01 steradians less than before then that's 0.03 Watts less of irradiance. But only if the light is hitting square on. If the handkerchief was rotated 60 degrees then it's only getting effectively illuminated at half intensity [cos(60 degrees) = 0.5]. So reducing its solid angle by 0.01 steradians will only reduce the irradiance by half as much, that is, by 3 * 0.01 * cos(60) = 0.015 Watts.

Note that this formula is either wrong or misinterpreted. Hard to tell without context. If you rotate your hanky by 60 degrees, the solid angle it subtends is already reduced by the factor of 0.5. There is no need to multiply by an additional factor of cos(theta). Doing so would be accounting for the same effect twice.

 

[Pilpellon]

It's from the book Real-Time Rendering.

Does it help understanding why the cos(theta) is there?

 

[jbriggs444]

Yes, that passage makes it clear that "irradiance" is measured per unit of surface area. There is a factor of cos(theta) ratio between incremental surface area and incremental subtended solid angle.