closing speed

[phyti]

This is in response to a recent thread on closing speed, which in my opinion was not totally resolved, primarily due to an inadequate definition of 'closing speed' and the improper application of the concept.

The original example is shown in fig. 1, with A and B moving at .7c toward M which is at rest relative to E.

The question is : What is the closing speed of A and B relative to E?

Typically closing speed is the rate of decreasing distance between two objects moving toward each other, but not necessarily in opposite directions. That would be a special 1-dimensional case and since this example is 2-dimensional, we consider all angles.

We define 'closing speed' as the rate of change of radial distance between two objects.

In fig. 2 the d-axis shows the radial distance from A (or B) to E, plotted as the hyperbola. It's obvious as A approaches M, the distance decreases to a minimum at M, then increases beyond M. The red curve crossing the 0-axis of E indicates the 'closing speed' of A relative to E. It approaches the limit .7c at extreme distances, but equals zero at M. Its value is negative for decreasing distance and positive for increasing distance. The dashed segments indicate a continuous path without collision at M. Only if there is no offset distance EM, will closing speed equal linear speed.

Relative to A, closing speeds are -.7c for M, and -.94c for B, per SR.

The collision of A and B separated by 1.4 ls happens in 1 sec because of simultaneous motions, not because of 'something' moving at 1.4c.

The linear and closing speeds of all objects involved have been calculated.

 

 

[swansont]

The closing speed is 1.4c. This is not at odds with relativity in any way.

[michel123456]

I don't understand fig.2.

 

((...) The red curve crossing the 0-axis of E indicates the 'closing speed' of A relative to E. It approaches the limit .7c at extreme distances, but equals zero at M. Its value is negative for decreasing distance and positive for increasing distance. The dashed segments indicate a continuous path without collision at M. Only if there is no offset distance EM, will closing speed equal linear speed.

(...)

 

In this example, by definition speed is constant. Speed does not equal zero at M. Distance does. IMHO. Or I don't read correctly.

Edited March 12, 2011 by michel123456

[phyti]

I don't understand fig.2.

 

 

 

In this example, by definition speed is constant. Speed does not equal zero at M. Distance does. IMHO. Or I don't read correctly.

 

As the green or red object passes M (at a constant speed), it's neither approaching nor receding from E, thus the closing speed of either relative to E is zero at that point. The closing speed varies because the path of the objects are offset from E. Closing speed for the objects relative to each other is constant because they are on a collision course.

[michel123456]

As the green or red object passes M (at a constant speed), it's neither approaching nor receding from E, thus the closing speed of either relative to E is zero at that point. The closing speed varies because the path of the objects are offset from E. Closing speed for the objects relative to each other is constant because they are on a collision course.

 

I hope I got it.

Labelling of both diagrams is unclear. In fig1, I understand vertical & horizontal axis are distances. In fig2. the upper part looks to have horizontal axis representing velocity, the down part as far as I understand velocity is the vertical axis.

Or I miss something.

Edited March 16, 2011 by michel123456

[phyti]

I hope I got it.

Labelling of both diagrams is unclear. In fig1, I understand vertical & horizontal axis are distances. In fig2. the upper part looks to have horizontal axis representing velocity, the down part as far as I understand velocity is the vertical axis.

Or I miss something.

You're right about inadequate labeling.

Fig. 2 is built on fig. 1 with M and E in the same orientation. Ignoring B, the upper curve (red) plots the distance AE. Coming from the left horizon it's decreasing. As A gets closer to M, AE decreases at a faster rate. As A passes M (the minumum separation) the distance AE is not changing. As A continues on (the green curve), the distance AE increases in the reverse manner. The horizontal axis through E plots the speed of A at each x location, which also equals the slope of the upper curve. Following the upper curve left to right, it slopes downward, levels off to zero at M, then increases,forming a mirror image of the left side. This should help you see the correspondence of the upper curve to the lower curve.

Edited March 17, 2011 by phyti