@Background Pony #E794  
Me too

It bugs me that the measurements are not proportionate with the visualization (to scale)…
 
_{Goddammit I hate that.}
 
I know. I bloody know it’s an equation and not really a picture, okay?
 
Calm your Angular gyrus.

I have a feeling Amber doesn’t actually know the physics behind this; she may have just took a general parabola and shifted some parameters around to set the question. This may also apply to the other question about Fluttershy and the mirror (>>850678).

Here is a formal solution. vy and vx are Pinkie’s vertical (right) and horizontal (up) velocities respectively while t1 and t2 are the times she reaches the zenith of her trajectory and the time she reaches the muffin. Standard gravity g does not play a part in the final solution, as will be demonstrated below, but may be taken to be -9.81.
 
Because we know that the zenith lies at 9 m and the muffin at 8 m lies along the trajectory, the following equations hold:
 
0 = vy + g * t1  
9 = 0 + vy * t1 + 1/2 * g * t1 ^ 2
 
Multiplying the first equation by t1 and subtracting from the second we get 9 = -1/2 * g * t1 ^ 2 and t1 ^ 2 = -18 / g; vy therefore equals -g * sqrt(-18 / g).
 
Substituting this into the second equation and expressing in terms of the original t1 we get:
 
8 = 0 + vy * t2 + 1/2 * g * t2 ^ 2  
1/2 * t2 ^ 2 - t1 * t2 + 4/9 * t1 ^ 2 = 0
 
and solving for t2 conveniently yields t2 = 4/3 * t1 (since t2 > t1, therefore the greater of the two solutions is relevant).
 
For the horizontal component, vx * t1 = 1/2 * L (the required distance) and vx * t2 = 4; dividing the first by the second and multiplying by 8 gives L = 8 * t1 / t2 = 8 / (4/3) = 6.
 
Therefore Pinkie and Twilight were separated by exactly six metres, regardless of how strong Equestria’s gravity is.

I would probably tackle it by:
 
using G=9.81m/s/s to calculate the time an object would take to decelerate to v=0m/s at s=9m and then calculate the time it would take to drop one meter (from 9 to 8).
 
This would give me the time that the object has taken to travel s=4m (horizontally), I can then calculate it’s horizontal speed V
 
I then go back again and calculate the time it takes for an object to start at s=0m decelerate to v=0m/s at s=9m and return to s=0m (This would be double the origonal calculation from my first part)  
This gives me total flight time.
 
Using total flight time and the horizontal velocity of the object, I can then easily calculate the total displacement.

bawhbb is right.  
Let’s do the full derivation  
the expression for an inverse parabole is f(x)= - ax² + bx + c  
if we let the start point at x = 0 be a 0 height we get that c = 0  
so f(x) = - ax² + bx  
we know that at x = 4 the height is 8, so - 16a + 4b - 8 = 0  
we’ll call this (1) and come back to that later  
we know that the parabola reaches a maximum at a height of 9  
to calculate the maximum of f(x) we take the derivative and equate it to 0  
df(x)/dx = - 2ax + b = 0 (to make sure it’s a maximum and not a minimum we can take the second derivative d²f(x)/dx² = - 2a, this must be negative to be a maximum, so a must be positive, as expected)  
this means the function reaches its maximum at x = b/2a  
since this must be where the height is 9 we get f(b/2a) = - b²/4a + b²/2a = 9  
or b²/4a = 9 or b = 6 sqrt(a)  
this relation between a and b can be inserted in (1)  
this gives - 16y² + 24y - 8 = 0 if we take y = sqrt(a) (shorter notation)  
this is a quadratic expression with as discriminant D = 24² - 4**(-16)**(8) = 64. this gives y = ( 24 - sqrt(64))/(-32) = 1 (and also 1/2, but that doesn’t apply here)  
therefore a = 1 and b = 6  
this makes our function f(x) = - x² + 6x = x(6 - x).  
Without needing to calculate a discriminant you can easily see this function will be 0 in x = 0 (which was a given) and x = 6 (our solution)

Here is the question, from the tumblr:
 
“There was a muffin held by Twilight’s magic 4 meters in front, 8 meters above Pinkie. So Pinkie jumped for the muffin. The route she went through was a parabola. She reached the highest point, which is 9 meters above the ground, and then she got the muffin. But when landing, she accidently landed on Twilight. So how far was Twilight in front of Pinkie so that the accident happened?”

There is no solution for x with only the information in the pic. Without speed or acceleration knowledge in x direction all the given info is worthless.

@Arianfis  
Eh, I’m rusty.

Let the lauch place be A, the top of the parabola be B, and the landing place be C.
 
From A to B, the muffin rises 9m, taking time t_1  
From B to C, the muffin falls 1m, taking time t_2  
Suppose the vertical velocity at A is v_y_A. We know v_y_A = gt_1  
Suppose the vertical velocity at B is v_y_B. We know v_y_B = 0
 
1/2 g(t_2)^2 = 1m  
((v_y_A)^2 - (v_y_B)^2)/(2g) = 9m  
Simplify,  
(t_2)^2 = 2m / g  
(t_1)^2 = 18m / g  
Thus t_1/t_2 = 3
 
The rest is obvious.

I’m going to make the assumption that Pinkie launches from the ground and lands at ground level, because otherwise this is impossible to solve.
 
The time to the top of the arc is 1.35s. Pinkie’s velocity at the 8m mark is 4.43 m/s. 13.28 m/s is her initial velocity in the y direction. So it takes her 1.8 sec to go 4m at 2.22 m/s. As such, at the midpoint of the arc, she’s gone 3 meters in the x direction, making your final answer x=6.
 
Nice problem.
 
@FanOfMostEverything  
It’s physics, not just quadratics :P

y=-x^2+6x
 
x=6

@FanOfMostEverything  
Curious, I got a simpler equation of:
 
y=-1x^2+6x
 
so x = Twilight = 5m
 
I didn’t even think about adding gravity to the mix. My old physics teacher would probably slap me on the back of the head.

6 m

@FanOfMostEverything  
..
 
b=41.2 cm?  
so.. x = (4-0.412) * 2

Actually, I may not. The parabola follows the form ax ² + bx + c. If we treat Pinkie’s pre-jump position as the origin, then c = 0. Assuming that a is acceleration caused by gravity and that Equestria’s gravity acceleration is that same as Earth’s (roughly -9.8 m/s ²), then:
 
-9.8x ² + bx = 8 where x = 4.  
-156.8 + 4b = 8  
4b = 164.8  
b = 41.2
 
So, to find x…
 
-9.8x ² + 41.2x = 0  
x = (-41.2 ± 41.2)/-19.6 = 0 or -82.4/-19.6, which is about 4.204 m. Assuming that the arc goes from Pinkie’s endpoint to Twilight’s midpoint, that means that ponies are about 40.8 cm long, and presumably not drawn to scale.

x=6

I’m going to need the equation of the parabola to determine x.

Those ponies are freaking huge..