Vector calculus sac help

[arthurk]

Jackson can be regarded as the line segment joining (-22,-22) and (-22,-20.2)
a) If Trish throws the can with a velocity 15m/s from (-32,-20.2) find 2 angles of projection that will cause Jackson to get hit.
b) Sketch the possible trajectory of the can found in part (a)
c) What is the minimum velocity that Trish needs to throw the can to make sure it hits any part of Jackson

Any ideas on how to do c?

[/0]

At the minimum velocity, the can should hit Jackson's feet.
Also, any angle of projection above the horizontal will require a lower minimum velocity than any angle below the horizontal.



From ,







For the horizontal motion,



In this formula, we can't simply change to make as negative as possible, because is also dependent on .

If we sub out though,



Now, we can minimize by picking , and hence minimize







In vector form this would be:

edit:nevermind

[arthurk]

Does the negative value for i indicate that she is throwing backwards?
Throwing at 225 degrees?

[/0]

Yes, negative value means she is throwing backwards.
Conventionally, angles are taken anticlockwise from the positive x-axis, so an angle of 225 degrees would correspond to the projectile being thrown to the left at 45 degrees above the horizontal.

[arthurk]

Am i wrong to say that since Trish is at (-32,-20.2) and the person shes throwing it at is at (-22,-22) then she must throw forward for it to hit him?

[arthurk]

And to hit his feet wouldn't the vertical displacement be -1.8 since his foot is at -22j and it is thrown from -20.2j

[/0]

yeah sorry I think I read it wrong

[/0]

I think it looks very messy...

For the vertical,

Using



Now in the horizontal,





We want to find the value of which will give the lowest value of ...

Since the function is implicit, perhaps
- Implicitly differentiating with being a function of , solving for 0 in terms of , then finding the corresponding .
- Using excel or graphing software (if allowed) to simply test many values of and .

[arthurk]

That looks epic
My ti-89 didn't let me differentiate it properly:(
Thanks anyway