Reference Frames And Relative Motion

The measured velocity of an object depends on the observer's reference frame. An observer moving in the same direction as the object measures a smaller velocity than a stationary observer. The magnitude and direction of the velocity both change with the choice of frame, because velocity is always measured relative to a specific coordinate system.

The claim is valid in the reference frame of the train. In the reference frame of a person standing on the ground outside, the book moves with the velocity of the train. Both descriptions are correct within their own frames.

Taking east as positive. Equation used: $${v}_{CT}={v}_{CG}+{v}_{GT}$$ The velocity of the ground relative to the truck is opposite to the truck’s velocity relative to the ground: $${v}_{GT}=-{v}_{TG}=-18\text{ m/s}$$ Given: $${v}_{CG}=+25\text{ m/s}$$ $${v}_{GT}=-18\text{ m/s}$$ Substituting: $${v}_{CT}=(+25)+(-18)$$ $${v}_{CT}=+7.0\text{ m/s}$$ The car moves east at 7.0 m/s as seen by the truck driver.

The velocity measured by the ground observer is greater than the ball's velocity relative to the bus. The ground observer's measurement combines the ball's velocity relative to the bus and the bus's velocity relative to the ground. Because both velocities point in the same direction, they add, producing a larger resultant.

Taking east as positive. Equation used: $${v}_{BG}={v}_{BT}+{v}_{TG}$$ Given: $${v}_{BT}=-5.0\text{ m/s}$$ $${v}_{TG}=+20\text{ m/s}$$ Substituting: $${v}_{BG}=(-5.0)+(+20)$$ $${v}_{BG}=+15\text{ m/s}$$ The ball still moves east relative to the ground, just slower than the train.

Each observer adds a different constant velocity when converting to their frame. This shifts all velocity values by the same amount. Because acceleration depends only on how velocity changes over time, and the shift is constant, the change in velocity over any time interval is identical for both observers. Therefore both observers calculate the same acceleration.

The ground observer measures an acceleration equal to 3.0 m/s². The train moves at constant velocity, so it is an inertial reference frame. Acceleration is the same in all inertial reference frames because only a constant velocity separates the two frames, and a constant does not affect the rate of change of velocity.