Find Components of Velocity U to Collide Particles & Minimize U

[philnow]

Homework Statement

The first part: find the x and y components of velocity U such that the particle fired with velocity U collides with the particle fired with velocity V.

The second part: for a given d, what should V be to minimize U?

The Attempt at a Solution

The first part is easy, I set t=v/g. This makes the x-component of U = d/t which = dg/v. Also, the y-component of U must be equal to V in order to collide with the V particle.

The second part is causing a lot of trouble for me. I understand that if V is small, the x-component of the U velocity will be too large, and if V is large, the y-component of U will be too large, so there must be some intermediate velocity V that makes U minimal. I'm just having trouble expressing this with equations... any help?

 

[kuruman]

Can you provide the statement of the problem as it is given to you? It is difficult to figure out what it is from what you have put down. Thanks.

 

[philnow]

You wish to simultaneously fire both particles so that they collide at their highest point. V is fired up vertically at the same time as U is fired.

The question as stated in textbook: what should U be (give the horizontal and vertical components). Given d, what should V be so that U is minimum?

 

[kuruman]

I see, I did not realize initially that gravity acts from top to bottom. For the second part, have you seen the so called "Monkey gun" demonstration? If not, go to



It should give you a clue about how to aim the projectile in order to hit the target.

 

[philnow]

That's very interesting lol but I don't really see it...

 

[kuruman]

As you said, the y-components must be the same to have a collision. When v has no y-component, then u must have no y-component. Anything that has only a x-component is smaller than anything that has both x and y components. So ...

 

[philnow]

I agree, but the question states that the particles collide at their peaks, so it's assumed that the speed V is greater than zero. Also, V should be in terms of, among other constants, the distance "d".

 

[kuruman]

Sorry, I missed that part. You know that d is half the range of the particle on the left and that the range is

$$R=\frac{2v_{0x}v_{0y}}{g}$$

Use this expression to find the x component of the velocity in terms of d and u. Once you have that you can get the speed v that you can then minimize.

 

[philnow]

D/2 = 2UxUy/g

D = UxUy/g

Ux = D*g/Uy
Ux = D*g/V

Well this is as far as I can get...

 

[kuruman]

You know Ux and Uy. Can you find the speed U?

 

[philnow]

U = sqrt(Uy^2 + Ux^2)
U = sqrt((t^2*v^2 + d^2)/t^2))

That's pretty ugly. If I use trig to find my U, I'll get cos(theta) = Ux/U

so U = Ux/cos(theta) or Ux = U*cos(theta)

this becomes U*cos(theta) = D*g/V

or U = D*g/Vcos(theta)

 

[kuruman]

How did time t get in the picture? You just said Ux = Dg/V and Uy = V. Just add the squares and take the square root. These are the components of U, the initial velocity of the projectile. Never mind what happens to them later.

 

[philnow]

Okay, U = sqrt((dg/v)^2 + v^2))

U = sqrt((d^2*g^2 + v^4)/v^2)

 

[kuruman]

Can you find the minimum of U with respect to V?