Abolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s^2. (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?

Refer to the diagrams shown below.
Define
a₁ = 2.5 m/s², acceleration of the car (north)
a₂ = the acceleration of the bolt relative to the car
g = 9.8 m/s²

Relative to the car, the acceleration of the bolt is
a₂² = 9.8² + 2.5² = 102.29
a₂ = 10.1 m/s²
The direction of a₂ is
θ = tan⁻¹ (2.5/9.8) = 14.3° south of the vertical.

Relative to the earth, the acceleration of the bolt is
a₂ = 10.1 m/s²
θ = 14.3°, north of the vertical.

(a) 10.1 m/s² at a vertical angle of 14.3° southward.
(b) 10.1 m/s² at a vertical angle of 14.3° northward