Tough kinematics question

1. Oct 31, 2003

joc

[POST CORRECTED]

would greatly appreciate it if someone could help me with the following problem:

2 point particles, A and B, are a distance d apart in a vacuum.

from time t=0 onwards, A has a constant nonzero velocity with magnitude v and a direction perpendicular to AoBo, where Ao and Bo are the initial positions of A and B.

from time t=0 onwards, B has a velocity of constant magnitude v which is always directed in direction BA (i.e. towards A).

find the distance AB at time=infinity.

(there are apparently relatively short ways to do this problem.)

Last edited: Nov 2, 2003
2. Oct 31, 2003

Guybrush Threepwood

it's probably d

3. Oct 31, 2003

StephenPrivitera

Hi joc! Welcome to the Forums.

You didn't show any work! Go to the homework section and read the Sticky post by Tom.

Since you're new here, I'll give you a hint, but next time show your work.
HINT:
It helps to know that both object are accelerating.

4. Oct 31, 2003

Ambitwistor

The problem states that neither object is accelerating.

5. Oct 31, 2003

Guybrush Threepwood

both have constant speed, both chage direction.... seems like acceleration to me

6. Oct 31, 2003

Ambitwistor

They have constant velocity: neither one of them changes speed or direction.

7. Oct 31, 2003

physics247

Apparently the speed of B is constant and its direction is changing as A moves. This is indeed acceleration.

Similarly, the magnitude of velocity of A is constant (ie speed is constant) but its direction is changing.

Last edited: Oct 31, 2003
8. Oct 31, 2003

Ambitwistor

Re: Re: tough kinematics question

Hmm, perhaps I misinterpreted the statement of the problem. I took "the line AB" to be the original line between the two particles.

9. Oct 31, 2003

physics247

In that case, they are both moving in a straight line, and at t = infinity they would be an infinite distance apart. Not much of a problem.

10. Oct 31, 2003

Ambitwistor

Yes, so I thought it was an odd problem... your interpretation is probably correct.

11. Oct 31, 2003

turin

If AB is considered to be the line that seperates the two particles, and A only moves perpendicular to this line (as stated in the original post), then the motion of B is the only thing that changes the distance between particles (because the motion of A is perpendicular to their separation, and therefore cannot effect the separation).

AB0 is the initial separation of A and B.

If B moves with a constant speed v directly towards A, then, at time t = AB0/v, the separation between the particles will be zero. If both AB0 and v are finite, then the particles will have zero seperation at time t = infinity (assuming that B always moves TOWARDS A).

I think the problem needs to be reworded.

12. Nov 1, 2003

HallsofIvy

Staff Emeritus
Yes that's true: "B has a velocity of constant magnitude v which is always directed in direction BA (i.e. towards A)."

The problem says "A has a constant nonzero velocity with magnitude v and a direction perpendicular to the line AB." Since it specifically says "constant nonzero velocity" I would interpret that to mean A travels along a straight line perpendicular to the line between the ORIGINAL positions of A and B.

13. Nov 1, 2003

Staff: Mentor

I think its poorly worded (maybe on purpose) but here's what appears to be going on:

A is moving at t=0 perpendicular to the line AB and continues in that direction with constant velocity (no change in speed or direction).

B is doing a "tail chase" which takes it on a curved (hyperbolic I think) path which by time infinity approaches being directly behind A.

You could construct an equation for the velocity component on B perpendicular to the original line AB, integrate for distance and subtract the distance A has traveled to find out how far apart they are.

14. Nov 1, 2003

jcsd

The line AA0 (A is A's current postion, A0 is A's intial psotion) is always vt, B's sepration from A (AB) is equal to [A0B0x2 + (AA0 - y)2]1/2 (x is B's component of separation from B0 in the direction parallel to A0B0 and y is B's component of separation from B0 in the direction paralell to AA0.

Due to the nature of the curve at t = infinity x = A0B0 and AB = y => (from the equations in the last paragraph)

AB = vt/2, which is an infinte separation.

Anyway that's my first attempt, I think though that I've either made a mistake or a faulty assumption (partly becasue I was expecting the answer to be finite and because I haven't really taken into account the fact that B's magnitude of velocity is always v along AB).

Last edited: Nov 1, 2003
15. Nov 2, 2003

krab

OK. I'll generously take the interpretation of this sloppily-worded problem that will make it the most interesting: both particles have velocity of fixed magnitude, but varying direction.

Particles A and B are initially side-by-side. Take a small time step. B has moved a little toward A, A has moved a little up, thus slightly tilting the line joining the two. So tilt your perspective a bit so the line is no longer tilted. So we are still in the same condition as at t=0, except the particles are a little closer together. Obviously then, at t=infinity, the distance AB is zero.

16. Nov 2, 2003

joc

my apologies to all. made a mistake in the wording; the question is actually as russ_watters and HallsofIvy have explained it:

A has a constant velocity (no change in magnitude or direction) which is directed perpendicular to the AoBo (the line segment AB at t=0).

B has a varying velocity with constant magnitude that is directed along line segment BA, B and A representing, for lack of a simpler phrasing, the current positions of the particles.

my attempts at working haven't come close to yielding any insights or answers so i didn't post them.

anyway, the answer is finite. btw russ, i've tried doing as you suggested but i have to involve an angle in order to obtain the components of velocity; i can't seem to get rid of the angle by expressing it in terms of time. seems like differential equations hold the answer...

17. Nov 2, 2003

Ambitwistor

18. Nov 2, 2003

turin

If the magnitude of the velocity of A is greater than or equal to the magnitude of the velocity of B, then B will never catch A.

19. Nov 3, 2003

joc

turin: the magnitudes are constant and equal. i'm looking for the distance between the 2 at time=infinity.

20. Nov 3, 2003

turin

OK, I think you will need to solve the differential equation.

Let the initial position of A be (xAo,yAo) = (0,0).

Let the initial position of B be (xBo,yBo) = -(AoBo,0).

The equation of motion for A is simply: (xA,yA) = (0,vt), where v is the magnitude of the velocity.

B is a little bit more complicated.
There are two equations that come to my mind: (dxB/dt)^2 + (dyB/dt)^2 = v^2,
and
(dxB/dt)/(dyB/dt) = (yA - yB)/(xA - xB)

Last edited: Nov 3, 2003