**Problem1A-3*:** Proof of Addition of Two Speeds and Two Accelerations

A train is moving toward the right as shown in the picture. A Person P is moving on the train. From the point of view of a ground based observer, the speed of the person P is the addition of the speed of the train and the relative speed of the person to the train. Similar addition of the acceleration of the train and the acceleration of the person relative to the train will result in the acceleration of the person viewed from the ground observer. Prove those statements.

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Let the ground observer be standing at the origin O. Denote the end point of the train as A, and the projection of A to the ground as point B. The projection of the position of the person to the ground is denoted as R. The distance between O and B at time t is denoted as f(t), and the distance between B and R at time t is denoted as g(t). Thus the distance between the observer and the person, that is, the distance between O and R at time t is f(t)+g(t). Thus the speed of the person v(t) and the acceleration of the person a(t) from the view of the ground observer are

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