Rank velocities of projectiles at landing?

[isukatphysics69]

Homework Statement

I will attatch the picture. Can someone please help me UNDERSTAND how to rank the velocity vectors at the landing zone?? I am having a really hard time with physics >=[ i don't know how to just look at this graph and rank the velocities.

Homework Equations

The Attempt at a Solution

I was using dummy variables for angles/components/time but realized that's not a good idea. Please someone help me understand how to approach this? i have a test in 2 days!

 

[PeroK]

This looks like an unusual question. If you fire a projectile (at any given angle), what happens to its trajectory as you increase its initial speed?

 

[isukatphysics69]

This looks like an unusual question. If you fire a projectile (at any given angle), what happens to its trajectory as you increase its initial speed?

The distance that the projectile travels increases right? But all of these distances are the same. I feel like since A will have the most time decreasing since it reaches its maximum before B, A>B. Since D is constantly decreasing i was thinking the magnitude of its vector will be greater than C at landing. i ranked it A>B?D>C and it was wrong.

 

[PeroK]

The distance that the projectile travels increases right?

Yes, but what about the shape of the trajectory? Perhaps think about trajectory B in your diagram, as an example.

 

[haruspex]

Clearly D>C and A>B, but no further comparisons are possible without making some estimate of the width to height ratio.
Let the width be x and the heights be y and 2y. For each case you will find $v^2=gy\left(\alpha+\beta\frac{x^2}{y^2}\right)$, for some constants α, β. Only by knowing an approximate value for x/y can you settle the whole sequence.