# Colliding particles

1. Sep 22, 2009

### philnow

1. The problem statement, all variables and given/known data

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?

3. 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?

2. Sep 22, 2009

### 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.

3. Sep 22, 2009

### 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?

4. Sep 22, 2009

### 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.

Last edited by a moderator: Sep 25, 2014
5. Sep 22, 2009

### philnow

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

6. Sep 22, 2009

### 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 ...

7. Sep 22, 2009

### 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".

8. Sep 23, 2009

### 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.

9. Sep 23, 2009

### 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...

10. Sep 23, 2009

### kuruman

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

11. Sep 23, 2009

### 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)

12. Sep 23, 2009

### 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.

13. Sep 23, 2009

### philnow

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

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

Last edited: Sep 23, 2009
14. Sep 23, 2009

### kuruman

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