Complete Square + Leibniz question (2 questions)

[Craig Scott]

Homework Statement

1. How did they complete the square for these equations in the picture below? What was their thought process?

2. distance/velocity = time , velocity/acceleration = time , In leibniz notation how does this cancel out?

When you divide, how does it cancel out to give you a time unit?

Homework Equations

1. In picture

2. ds/dt / d²s / dt²

The Attempt at a Solution

1. It seems like it came out of thin air.

2. dt/ds = time?

 

[BiGyElLoWhAt]

What is 2. in your relevant equations?

$V = d/t \to t = d/V$
You can add A and subtract A (equivalent to adding 0), and it's still the same.
$\frac{v_{y0}x}{v_{x0}} - \frac{gx^2}{2v_{x0}^2} + \frac{v_{y0}^2}{2g} - \frac{v_{y0}^2}{2g}$
They factored out $\frac{-g}{2v_{x0}^2}$

Rearrange it and it might look a little closer.

 

[Craig Scott]

In my relevant solutions it is velocity/acceleration, the way I put it was just in leibniz notation in terms of s(t)

 

[BiGyElLoWhAt]

I think I was just confused by all the slashes. Sort of makes it look like one big fraction made of fractions.

 

[Craig Scott]

V²_y0/2g

How did you find that

 

[haruspex]

V²_y0/2g

How did you find that

I assume you are asking about getting from the second equation in the image to the third.
Just expand the square in the third equation and simplify. You should arrive at the second equation.