# Projectile Motion Using Vectors

• themethetion
In summary: to shorten the range to the blue curve while maintaining the point ##(x_1,y_1)## you would have to increase both the velocity and the angle?
Is this some math exercise or do you want it to be of any actual use for tennis practice? Drag and Magnus effect due to ball spin are not negligible in the real world.

FranzS said:
Is this some math exercise or do you want it to be of any actual use for tennis practice? Drag and Magnus effect due to ball spin are not negligible in the real world.
Not to mention that a lob may be hit from well behind the baseline and that most players don't remain statuesque at the net, but run backwards to hit a smash.

PeroK said:
... most players don't remain statuesque at the net ...
... with their racquets raised up high like a present-day Statue of Liberty ...

FranzS said:
Is this some math exercise or do you want it to be of any actual use for tennis practice? Drag and Magnus effect due to ball spin are not negligible in the real world.
PeroK said:
Not to mention that a lob may be hit from well behind the baseline and that most players don't remain statuesque at the net, but run backwards to hit a smash.
It's a math exercise that we must make as practical as possible to 'help inform players in performing a successful shot'.

themethetion said:
TL;DR Summary: Using vector functions how can I find the minimum average velocity (something greater than 11.86m/s) of a ball if the launch angle is unknown and if I have a point that the object must travel through (11.86, 3.47)?
Not sure about vector "functions" as such but you can use:

$$\vec{a}\times\vec{s} = \vec{v} \times \vec {u}$$

As @kuruman has elsewhere advised, minimum velocity corresponds to ##\vec{v}## and ##\vec{u}## being perpendicular. In this case, the above simplifies to:

$$gR = u\sqrt{u^2-2gh}$$ where R is the given x-coordinate and h the given y-coordinate.

Edit y-coordinate should be less one given the ball is hit from 1m above ground. See post #4.

https://www.desmos.com/calculator/3byo9fl9ha

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