A particle P2 chases particle P1 with constant speed.

[Ishan Sharma]

Homework Statement

[/B]
A particle P1 moves with a constant velocity v along x-axis, starting from origin. Another particle
P2 chases particle P1
with constant speed v, starting from the point (0, d). Both motion begin
simultaneously. Find -

1. Initial acceleration of P2.
2. Ultimate Separation between P1 and P2
3. Whether the trajectory of P2 in frame of P1 will be Hyperbola or Parabola.

The Attempt at a Solution

Let r be the distance between 1 and 2 at any
time t.
$$vcosθ- v= \frac{dr}
{dt} ... (1) $$
$$vcos\theta= \frac{dx}
{dt} ... (2)$$

For the rest of solution, please refer the attachment.

My questions-

1. This ω, according to me is the rate of change of direction of velocity vector. Now, I couldn't think about dθ/dt directly. So, what I did was, I saw took the ω with respect to P1. Is that right? If not, what's the correct answer?
2. I don't know what to do with the trajectory.

Any help would be appreciated.

Thankyou.

 

[kuruman]

You may find this link useful.
http://mathworld.wolfram.com/PursuitCurve.html

 

[haruspex]

I am impressed by the elegance of your solution to the first part.
For ω, not sure what your difficulty is. The rate of rotation will be the same in all inertial frames.
For the trajectory, how does the relative x position relate to the terms in your dx=vdt+dr?

 

[Ishan Sharma]

For ω, not sure what your difficulty is. The rate of rotation will be the same in all inertial frames.
For the trajectory, how does the relative x position relate to the terms in your dx=vdt+dr?

I am not sure about ω because the two particle system can not be treated as a rigid body. So how can we say that ω with respect to (w.r.t) ground frame will be same as that w.r.t. any inertial frame?
And for trajectory, can we not give some qualitative or intuitive argument without forming the actual equation..? For eg: If we show that the distance of a particle from a fixed point is constant, we can say it moves in a circle. In case of conics, their definition can be used to find the nature of path. Can we say something similar about our question..?

 

[haruspex]

I am not sure about ω because the two particle system can not be treated as a rigid body. So how can we say that ω with respect to (w.r.t) ground frame will be same as that w.r.t. any inertial frame?

Yes, rate of rotation is the same in any inertial frame. Indeed, an accelerating frame is ok, as long as it is not itself rotating.

And for trajectory, can we not give some qualitative or intuitive argument without forming the actual equation

You can give a geometric argument. After answering my question in post #3, look at https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2a/2a_2content_10.html.

 

[Ishan Sharma]

r+Δx=d
⇒r=d-Δx
Thus the distance between focus and directrix is d and the distance between particles is always equal to distance between P2 and directrix. Thus the path is parabolic. Am I right, sir?

 

[haruspex]

r+Δx=d
⇒r=d-Δx
Thus the distance between focus and directrix is d and the distance between particles is always equal to distance between P2 and directrix. Thus the path is parabolic. Am I right, sir?

Yes.