Solving Physics Problems (Part 1)

So I went to Powell's books in downtown Portland and I found a book by Leo Sartori entitled Understanding Relativity: A simplified Approach to Einstein's Theories.  This book deals with Einstein's theory of special relativity.  I've been enthused by the book to try doing the problems at the end of the chapters and I will now share my work with you, dear loyal reader.

Problem 1.1

A train moves at a constant speed.  A stone on the train is released from rest.
(a) using the principle of relativity, describe the motion of the stone as seen by observers on the train.
Answer: As seen by observers on the train, the stone, having the same initial velocity as the train, fall straight down.
(b) Using the Galilean transformation, describe the motion of the stone as seen by observers on the ground.  Draw a sketch.
Answer: To the observer on the ground, the train moves to the right with a speed V.  Therefore, the stone also moves to the right with a speed V as it falls.  See the sketch below:

Problem 1.2

Let V = 30 m/s, h₀ = 7.2 m, approximate g as 10 m/s.

(a) Write the equations that describe the stone's motion in frame S'.

Answer:

S': x' = 0 ; y' = h₀ - ½gt'² ; z' = 0

(b) Use the Galilean transformation to write the equations that describe the position of the stone in frame S.  Plot the curve of the motion of the stone in frame S.

Answer:

The transform from S' → S is as follows:

x = x' + Vt'

y = y'

z = z'

t = t'

The equations of motion in S are:

S: x = Vt ; y = h₀ - ½gt² ; z = 0

Here is the plot of the motion (from Desmos)

(c) Write the equations for the three components of the stone's velocity in S' and use the Galilean transformation to find the component in S.
Answer:
The components of the stone's velocity in S' are:

The Galilean transformation is:

And the components of the stone's velocity in S are:

(d) Find the magnitude of the stone speed in each frame at t = 1 sec.

Answer:

The magnitude of the velocity vector in frame S and S' are: