Relative motion of a dropping object on a moving train

[MechaMZ]

Homework Statement

A bolt drops from the ceiling of a train car that is accelerating northward at a rate of 3.50 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?

The Attempt at a Solution

(a)10.4 m/s² at 19.7° to the south from the vertical
(b)9.8 m/s² vertically downward


I don't understand why the acceleration in (a) is not vertically downward since it is relative to the train. As we could assume the train is not moving, then the motion should be vertically downward isn't?

 

[Jebus_Chris]

The train is accelerating. So the instant the bolt starts to drop you would see it going vertically downward if you were on the train. Now since the train is accelerating the bolt appears to be moving backwards since the train is speeding up but the bolt is not.

 

[tomcue]

I would like to clarify that the acceleration in (a) is not vertically downward because it is relative to the train, which is accelerating northward. This means that the bolt is experiencing two types of motion simultaneously - its own downward motion due to gravity, and the train's northward acceleration. These two motions combine to create a resultant acceleration that is at an angle to the vertical. This is known as the "vector sum" of the two accelerations.

To better understand this concept, imagine standing on a moving train and dropping a ball straight down. To you, the ball appears to fall straight down, but to someone standing outside the train, the ball appears to fall at an angle due to the train's motion. This is because the ball is also moving with the train and its motion is affected by the train's acceleration.

In the case of the bolt dropping from the ceiling of the train, it is also affected by the train's acceleration and therefore has a resultant acceleration at an angle to the vertical. This does not mean that the bolt is not falling vertically, it simply means that its motion is affected by the train's acceleration.

In contrast, the acceleration in (b) is relative to the Earth, which is not accelerating in this scenario. Therefore, the bolt experiences only its own downward motion due to gravity, resulting in a vertical acceleration of 9.8 m/s2.

In summary, the relative motion of a dropping object on a moving train is affected by both the object's own motion and the train's motion, resulting in a resultant acceleration that is at an angle to the vertical. This is a concept known as "vector addition" and is commonly used in physics to describe the motion of objects in different reference frames.