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Video: Galilean Transformation of Velocity

Ed Burdette

A man standing still at a train station watches a train move past him eastward at 20 m/s. Two boys inside the train throw a baseball. The ball is thrown westward, at a speed of 5.0 m/s relative to the boy that throws the ball. What is the velocity of the ball as measured by the man in the train station? Assume that eastward corresponds to positive values of displacement.

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Video Transcript

A man standing still at a train station watches a train move past him eastward at 20 meters per second. Two boys inside the train throw a baseball. The ball is thrown westward at a speed of 5.0 meters per second relative to the boy that throws the ball. What is the velocity of the ball as measured by the man in the train station? Assume that eastward corresponds to positive values of displacement.

In this statement, we’re told that the train is moving eastward at a speed of twenty meters per second. We’ll call that 𝑣 sub t, the velocity of the train which is 20 meters per second to the east. We’re also told that inside the train, a ball is thrown at a relative speed of 5.0 meters per second to the west. We’ll call that 𝑣 sub b, the velocity of the ball relative to the train which is 5.0 meters per second west. We want to solve for the velocity of the ball in the man’s frame of reference. We’ll call that 𝑣 sub m.

To begin, let’s draw a diagram of the scenario. In this diagram, the man standing on the train platform sees the train move past him to the east moving at a speed of 𝑣 sub t. On board the train, a boy throws a ball in the opposite direction moving at 𝑣 sub b, 5.0 meters per second to the west.

Defining motion to the east as positive motion, we can rewrite 𝑣 sub b as negative 5.0 meters per second to the east. 𝑣 sub m is the velocity of the thrown ball relative to the man on the platform. We find that velocity by adding together 𝑣 sub t and 𝑣 sub b. Again, defining motion to the east as positive motion, 𝑣 sub t is equal to positive 20 meters per second and 𝑣 sub b is equal to negative 5.0 meters per second.

Adding these values together, to two significant figures, the velocity of the ball relative to the man, 𝑣 sub m, is 15 meters per second. This is how fast the man on the platform sees the ball moving.