A man standing still at a train station watches two boys throwing a baseball in a moving train. Suppose the train is moving east with a constant speed of \(20 \mathrm{m} / \mathrm{s}\) and one of the boys throws the ball with a speed of \(5 \mathrm{m} / \mathrm{s}\) with respect to himself toward the other boy, who is 5 m west from him. What is the velocity of the ball as observed by the man on the station?

Understanding relative motion is pivotal for solving problems in physics, particularly when objects are in motion with respect to different observers or reference points. Imagine you're on a train, and you toss an apple up into the air. To you, the apple moves straight up and down. However, to a person standing still outside the train, the apple travels in a curved path because the train is moving forward.

Similarly, in the problem given, there are two observers: the man standing still at the train station and the boy in the moving train. The ball's motion appears different to each observer because the ball's velocity is relative to the observer’s frame of reference. The man sees the ball moving in conjunction with the train's motion, unlike the boy on the train, who perceives the ball to be moving in the opposite direction of the train's movement. This discrepancy in observation is an example of relative motion at play where the velocity of an object varies based on the observer's frame of reference.

Imagine watching the world from a camera lens; the scene you capture depends on your position and orientation. Similarly, a frame of reference in physics is like a vantage point with its own set of axes to measure positions and velocities of objects in relation to it. It’s the 'coordinate system' for an observer, helping them describe how objects move.

Frames of reference can be stationary or moving. In our problem, we have two frames of reference: the train (a moving frame of reference) and the station (a stationary frame of reference). Each observer uses their own frame to describe the ball's motion. For the boy on the train, the frame of reference is moving with the train, while the man at the station uses a frame of reference that is fixed to the ground.

To quantify observations accurately, it's essential to specify the frame of reference. This is especially important when dealing with relative velocities; measuring speed and direction must be tied to a specific frame to have any real meaning.

Why do vectors matter? Because they let us combine multiple motions into one. A vector represents both magnitude (how much) and direction (which way). Vector addition is how we combine these quantities, especially when dealing with velocities in more than one dimension or in different directions.

In our exercise scenario, there are two vectors to add: the velocity of the train and the velocity of the ball as thrown by the boy. When calculating the velocity of the ball as seen by the man at the station, we need to add these vectors taking both magnitude and direction into account. This is crucial because simply adding their speeds wouldn't give us the right answer, direction has to be considered.

Visualize each velocity as an arrow pointing in its direction: the train's vector points towards the east, the ball's towards the west from the boy's perspective. When the boy throws the ball, its relative velocity vector points in the opposite direction of the train's movement. To find the ball's velocity from the man’s viewpoint, we flip the ball's vector to align with the train’s direction for the man's frame of reference, and we add the numerical values. This is why, even though the ball is thrown westward inside the train, the resulting vector shows eastward movement to the stationary observer.