# B Relative motion of converging objects

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1. Dec 29, 2017

### JulianM

If two objects are travelling on paths at 90 deg to each other such that they will converge at a point how do we describe that motion.

For example 2 cars are approaching a 4 way intersection and will collide. They see each other as moving diagonally yet when they collide one will t-bone the other at 90 deg.

2. Dec 29, 2017

### PeroK

What about "on a collision course"?

3. Dec 29, 2017

### JulianM

Ha Ha, yes, but I meant in terms of Physics. Are they moving diagonally ? If so why do they intersect orthogonally.

4. Dec 29, 2017

### Nidum

Do They ?

5. Dec 29, 2017

### PeroK

Angle is dependent on reference frame. When you say diagonally, you are thinking of the problem in the reference frame of one of the cars. And when you say orthogonally, you are thinking in terms of the ground reference frame.

Idea: Consider not a collision, but a near miss. Draw this from the frame of one of the cars and in the ground frame.

6. Dec 29, 2017

### JulianM

Well I am not sure I know.

If one car is half a car length ahead then the other will t-bone it square in the side, yet when they consider each other the were moving diagonally weren't they?

7. Dec 29, 2017

### JulianM

When I say orthogonally what I mean is, if one arrives very slightly ahead of the other, or even a narrow miss, they will be at 90 deg to each other.

I don't want to get out of my depth on this, but we have been taught that there is only relative motion and no ground frame, but I am very, very confused.

8. Dec 29, 2017

### PeroK

When someone says "all motion is relative", you have to understand the context in which that is meant. What it means is that you cannot assign a definite, absolute velocity to anything. In your example of the cars, someone might say that one car is travelling at $90km/h$. But that is relative to the road. The road is on the surface of the Earth, which is spinning at $1000km/hr$ and the Earth itself is orbiting the Sun at whatever speed.

It also means that when two objects are in relative motion, you cannot say absolutely that one is moving and the other is not. In each object's rest frame it is the other object that is moving. And, in a third frame, both objects may be moving.

So, your car has no definite, asbolute velocity. Only a velocity in a given frame of reference. The ground frame, which is perfectly valid by the way, is as good a frame as any. In fact, we use it all the time. Game of football, tennis, whatever. The obvious reference frame to study a tennis match is in the ground frame!

Hopefully that explains that.

Back to your problem, which is a good question in fact. There are three obvious frames of reference in which to study this problem: the rest frame of each car; and the ground frame.

To repeat, angles and trajectories are frame dependent. If you were a traffic controller, then your frame would no doubt be the ground frame. Let's assume car A is heading East/right along the x-axis and car B is heading North/up along the y-axis. These paths are clearly at right angles to each other.

If you are in car A and use your rest frame to analyse the problem, then car B is coming at you from the bottom-right (South-East). In your rest frame, therefore, car B is not moving north up the y-axis, but at an angle, somewhere is the NW direction.

One key point to note is that if there is a collision, then it doesn't matter which frame you use, the results of the crash (if you correctly apply the laws of physics) will be the same.

9. Dec 29, 2017

### JulianM

Thank you. Very helpful. My follow then is i presume the correct law to apply is that the force is diagonal.

10. Dec 29, 2017

### PeroK

The direction of a force is, you know what I'm going to say, frame dependent!

Let's assume a totally inelastic collision: that means the two cars get stuck together after the crash. Let's also assume that the cars have the same mass and, in the ground frame, the same speed.

In the frame of car A, all the momentum is in the NW direction. The cars, therefore, will move off in this direction at a given speed, call it $v$, after the collision.

In the ground frame, the momentum is NE, so the cars will move off at a speed of $v$ in the NE direction.

The point is that these two analyses represent the same physical result. For example, if there is a road running NE in the gorund frame, then the cars will move up that road after the collision.

You may need to draw a diagram to see that they also move up that road when you analyse the problem from the frame of car A.

11. Dec 29, 2017

### JulianM

Thank you, sir.

12. Dec 29, 2017

### jbriggs444

Four. The center of mass frame is a very useful choice.