A bolt drops from the ceiling of a train car that is accelerating northward at a rate of 2.15 m/s2.

(a) What is the acceleration of the bolt relative to the train car?

(b) What is the acceleration of the bolt relative to the Earth?

As soon as the bolt detaches from the ceiling, its northward acceleration

stops, and its downward acceleration begins.

Relative to the accelerating train car, the bolt has acceleration of

2.15 m/s² south, and 9.8 m/s² down. When you puttum together,

the resultant of those two components is

√ (2.15² + 9.8²) = √100.6625) = 10.03 m/s²

in the direction of tan⁻¹ (2.15/9.8) = 12.4° south of straight down.

Relative to the Earth, the bolt has no horizontal acceleration and only vertical

acceleration of 9.8 m/s² straight down (toward the center of the Earth).

Note that I'm ignoring the centripetal acceleration toward the Earth's axis

due to daily rotation, and the centripetal acceleration toward the sun due

to the Earth's yearly revolution. These are both relatively small, compared

to the acceleration of the train or of gravity.