Problem 1C-06*: Addition of velocity and acceleration vectors

(a) A train is moving in a flat plane with a velocity vector v at certain moment as shown in Fig.1. A coin located at point P on the rough floor of the train is sliding with a velocity u relative to the train at that moment. From the view of a ground observer, what is the velocity vector of the coin?

(b) Suppose that the train is entering a curve at that moment and has an acceleration vector a as shown in Fig.1. On the other hand the coin is experiencing a deceleration that has a magnitude b and is antiparallel to u. What is the acceleration vector of the coin observed by the ground observer?

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Solution:

(a) Since the point of P of the floor is moving along with the train, it has a velocity vector w that is parallel to u, has the same magnitude as the magnitude of u, and originates from point P. The floor is rough, so the coin is pulled by point P and inherits the same velocity vector w. The velocity vector w of the coin can be summed with the velocity vector u of the coin and results in the velocity vector w+u that the ground observer observes as shown in Fig.2

(b) The same argument as part (a) indicates that the coin inherits an acceleration vector A that is parallel to a, has the same magnitude as the magnitude of a, and originates from the coin at point P at the moment of concern. The deceleration vactor of the coin, denoted as b, that also originates from the coin at point P is

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Make sure that the vector b defined above has a magnitude b and is antiparallel to u. The two vectors A and b then can be summed to form the acceleration vector A+b as shown in Fig.3. A+b is the acceleration vector of the coin that the ground observer observes.