# B How does calculus relate to dimensions?

1. Jan 7, 2018

### paulo84

I am trying to understand what time^2 and velocity^2 mean in terms of how to visualize them? This wasn't explained in Physics or Mechanics (Further Mathematics) in high school, unfortunately. It seems likely it relates to matrices, maybe?

Appreciate any replies! :)

2. Jan 7, 2018

### PetSounds

For acceleration, you're better to think of it as metres per second per second rather than metres per second squared. For example: In freefall, your speed increases at a rate of 9.8 m/s per second. Mathematically, this is the same thing as 9.8 m/s2, but it's a little easier to wrap your head around.

3. Jan 7, 2018

### paulo84

Thanks. It's just... a square is a shape? Or not necessarily?

4. Jan 7, 2018

### Matthew314159271828

There isn't any way to "visualize" why there is a velocity squared term in the kinetic energy equation. Most equations in physics are in a mathematically simplified form that doesn't reveal the laws and principles that were used to derive them. Moral: Understanding how the equation is derived is more valuable than trying to "visualize" its individual terms.

5. Jan 7, 2018

### PetSounds

To square a number means to multiply a number by itself. It doesn't always refer to a geometric shape.

6. Jan 7, 2018

### paulo84

But isn't the number we get from squaring only known because of that shape? Or not?

OK...I'm visualizing hidden squares! Folded back on themselves...I appreciate your point about the important part to understand, but do you think that would be correct??

7. Jan 7, 2018

### Staff: Mentor

I am assuming that you know the difference between vectors and scalars. Energy is a scalar and velocity is a vector, so if you want an energy which is related to velocity then you need some operation which takes a vector and returns a scalar.

That operation is known as the dot product. So $v^2$ is shorthand for $v \cdot v = v_x^2 + v_y^2 + v_z^2$. This is the square of the speed.

As far as visualization, it means that if the x axis is speed and the y axis is energy then the graph is a parabola with the vertex at the origin

8. Jan 7, 2018

### paulo84

Wow, thanks, that's awesome! Now I have some reading to do.

9. Jan 7, 2018

### Matthew314159271828

Look, there is no way to visualize what the equation is conveying in a physical sense. The mathematical visualization in terms of a graph is not important to the physicist.

10. Jan 7, 2018

### PetSounds

Why? You can multiply 3 × 3 without involving geometry.

11. Jan 7, 2018

### paulo84

Oh sorry, I should have asked the question in the Maths subforum.

12. Jan 7, 2018

### Staff: Mentor

I wouldn’t go that far. If graphs and visualizations weren’t important to physicists then there wouldn’t be so many of them in physics papers and textbooks.

13. Jan 7, 2018

### Matthew314159271828

You're absolutely right

14. Jan 7, 2018

### paulo84

Dale - I came back to your post. So the vector velocity is the sum of the speed along each of x, y and z axes?

Relative to acceleration, is there an implication that you're dealing with 2 dimensions of time, or not?

15. Jan 7, 2018

### paulo84

Sorry, mixing up my concepts. The square of velocity is the sum of the square of the speed along each of x, y and z axes?

16. Jan 7, 2018

### paulo84

And likewise velocity is essentially speed in 3 dimensions, which can be expressed as a matrix?

17. Jan 7, 2018

### Jayalk97

Look my dude, you're overcomplicating this in your head. Let's start from the bottom. Position is where you are. The rate at which you change position is velocity. The rate at which you change velocity is acceleration. Change in position is distance per second. We can denote that as meters per second, or meters/seconds. Acceleration as the rate at which your velocity changes. This could be denoted as velocity per second, or velocity/seconds. If you sub in the definition of meters per second into that you get meters/(second^2).

18. Jan 7, 2018

### Jayalk97

I should also include that velocity has two components, speed and direction. Depending on the number of physical spatial dimension you are working with this, the direction part can have multiple components, one for every direction you're able to move in.

Last edited by a moderator: Jan 7, 2018
19. Jan 7, 2018

### paulo84

Thanks. I need to look at it in 3D to understand it! I was told in school I was a natural mathematician and a natural chemist but not a natural physicist. I only made it through the Mechanics paper because I could work out the inbetween maths, I sometimes have trouble grasping concepts which seem simple to others...

20. Jan 7, 2018

Tensors?