Problem 1A-4*: Addition and Subtraction of Speeds

There is a train moving to the right with a constant speed of 9.0 km/hr as shown in the picture. On the train a person P walks in the direction of the moving train with a constant speed of 4.0 km/hr. His dog D, at the other end of the train, runs toward him with a constant speed of 6.0 km/hr. There are two observers, A and B, and both of them are dressed in red from head to toe. Observer A is standing still on the moving train and Observer B is standing still on the ground.

(a) From the viewpoint of Observer A, what are the speeds of the person P, the dog D, the observer B and the train respectively? Give both the magnitude and the direction for each speed.

(b) From the viewpoint of Observer B, what are the speeds of the person P, the dog D, the observer A and the train respectively? Give both the magnitude and the direction for each speed.

(c) From the viewpoint of the person P what are the speeds of the dog D,the observer A, the observer B and the train respectively? Give both the magnitude and the direction for each speed.

(d) Suppose that the ground observer suddenly starts to walk in the direction of the train with an acceleration of
1 m/sec2 until he reaches a speed of 3.0 km/hr. Then he just walks with that speed constantly.
Describe the movement of the dog from the viewpoint of the ground observer.

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Solution:

(a) From the viewpoint of Observer A, he is not moving, but everyone else is moving. Thus:
Person P is moving toward the head of the train with a speed of 4.0 km/hr,
Dog D is moving toward the rear end of the train with a speed of 6.0 km/hr,
Observer B is moving toward the rear end of the train with a speed of 9.0 km/hr,
and the train is not moving.

(b) From the viewpoint of Observer B, he is not moving and the train and everyone else is
moving with the train. Thus:
Person P is moving in the direction of the moving train with a speed of 4.0 + 9.0 = 13.0 km/hr,
Dog D is moving in the direction of the moving train with a speed of 9.0 - 6.0 = 3.0 km/hr,
Observer A is moving in the direction of the moving train with a speed of 9.0 km/hr,
and the train is moving with the speed of 9.0 km/hr.

(c) From the viewpoint of Person P, he is not moving. Thus:
The dog is moving toward him with a speed of 4.0 + 6.0 = 10.0 km/hr,
Observer A is moving away from his back with a speed of 4.0 km/hr,
Observer B is moving toward the back with a speed of 4.0 + 9.0 = 13.0 km/hr,
and the train is sliding back under his feet with a speed of 4.0 km/hr.

(d) Before the ground observer started to walk the dog was moving toward the right with a speed of 3.0 km/hr. Then the
observer will see that the dog started to accelerate toward the left with a magnitude of 1.0 m/sec2. That acceleration
toward the left slows down the the dog's rightward movement. Eventually the dog becomes stationary from the view of
the ground observer. During this whole process the legs of the dog are constantly moving. But the floor of the train
accelerates toward the left first and then eases back to a constant speed of 9.0-3.0=6.0 km/hr. The dog is like a person
walking on a varying speed treadmill; the legs move but do not need to go anywhere.