Why Does Vqp Need to be Tangential to Vq in Order for Q to Get Closest to P?

[Jas]

https://www.physicsforums.com/attachments/221718 Say if we have two particles P, and Q, traveling at velocities Vp and Vq respectively. If it is IMPOSSIBLE for Q to collide with P, let us find the distance of closest approach. So from the frame of reference of P, itself is stationary, and Q is moving at Vqp (velocity of q relative to p). In order to get the closest to P, the velocity of Q (Vq) can take a locus of velocities, forming the shape of a circle of radius mod(Vq).

Why is it that in order for Q to get the CLOSEST to P, Vqp has to be tangential to Vq from the circle?This is in 2 dimensions, and by collide, I mean intercept

 

[jbriggs444]

Say if we have two particles P, and Q, traveling at velocities Vp and Vq respectively. If it is IMPOSSIBLE for Q to collide with P, let us find the distance of closest approach. So from the frame of reference of P, itself is stationary, and Q is moving at Vqp (velocity of q relative to p). In order to get the closest to P, the velocity of Q (Vq) can take a locus of velocities, forming the shape of a circle of radius mod(Vq).

Why is it that in order for Q to get the CLOSEST to P, Vqp has to be tangential to Vq from the circle?This is in 2 dimensions, and by collide, I mean intercept

Without a diagram, you've certainly lost me.

Since you are talking about circles, you must be talking about planar motion. If you are mention a locus of velocities forming a circle, you must be talking about a fixed relative speed with an unknown direction. But in order to get "closest" to P, the relative velocity of Q must simply be antiparallel to the displacement of Q from P. No mystery and no tangents there.

If you want to hit something that's motionless: aim directly at it.

Edit: I see that kuruman has assumed that this is an intercept problem. You are launching a projectile from Q at P with some fixed speed (relative to Q - a bullet?) (relative to an absolute frame - a torpedo?)

 

[kuruman]

Why is it that in order for Q to get the CLOSEST to P, Vqp has to be tangential to Vq from the circle?

Because the closest distance between a point and a straight line is the perpendicular from the point to the line. Furthermore, the tangent to a circle is perpendicular to the radius. So the perpendicular from the point to the line is the radius of the circle.

On edit: The radius of that circle is also called the impact parameter, look it up.

 

[Jas]

Without a diagram, you've certainly lost me.

Since you are talking about circles, you must be talking about planar motion. If you are mention a locus of velocities forming a circle, you must be talking about a fixed relative speed with an unknown direction. But in order to get "closest" to P, the relative velocity of Q must simply be antiparallel to the displacement of Q from P. No mystery and no tangents there.

If you want to hit something that's motionless: aim directly at it.

Edit: I see that kuruman has assumed that this is an intercept problem. You are launching a projectile from Q at P with some fixed speed (relative to Q - a bullet?) (relative to an absolute frame - a torpedo?)

I've added a diagram. As you can see, qVp (same thing as Vqp) is tangential to the locus of Vq

 

[Jas]

Because the closest distance between a point and a straight line is the perpendicular from the point to the line. Furthermore, the tangent to a circle is perpendicular to the radius. So the perpendicular from the point to the line is the radius of the circle.

On edit: The radius of that circle is also called the impact parameter, look it up.

But we're talking about the tangent between two velocity vectors

 

[kuruman]

But we're talking about the tangent between two velocity vectors

I am not sure what this is. Anyway, posting the problem that relates to your question clarifies the situation. I will have to think about the answer to your question, but now I have to sign off for a few hours. Perhaps someone else will be able to help you.