Passeneger train applied brakes and decelerates at uniform rate?

Question

If a passenger drops golf ball and ping-pong ball from the same heigh, which ball will fall further from the passneger?

The ping-pong ball will fall slower, of course, due to air drag.

Answer 1

Let v = velocity of train = v0 – a*t
We assume that the air inside the train has the same instantaneous velocity as the train.
Let ux and uy be the horizontal and vertical components of the absolute velocity of the ball.

Let’s assume that the drag force is proportional to the first power of velocity, although I believe the analysis will hold for any function.

Horizontally:
m*dux/dt + k*ux = k*v0 – kat
The general solution, accounting for BCs, is
ux(t) – [v0 – at] = ma/k * [1 - exp(-k*t/m) ]
Integrating
x(t) – [v0*t – ½ at^2] = ma/k * [t + m/k * exp(-kt/m) – m/k]
= relative distance between ball and person as a function of time

Vertically:
m*duy/dt + k*uy = mg
The general solution is:
uy(t) = mg/k * [1 – exp(-kt/m]
Integrating
y(t) = mg/k * [t + m/k * exp(-kt/m) – m/k]
where y is measured downward from initial position of ball

Let’s suppose the ball is dropped from a distance h above the ground
y = h when
t + m/k * exp(-kt/m) – m/k = kh / mg
and x_rel = ma/k ( kh/mg) = ah/g

This measures the distance the ball drops from the person, and is independent of the mass of the ball. This means that both balls hit the ground at the same spot, although not at the same time.

Answer 2

They will both fall the same distance away.

The reason I think the question involves a golf ball and a ping pong ball is because they both are approximately the same size and shape, and so will experience the same air resistance. Therefore air resistance is not a consideration here.

And although the golf ball is heavier than the ping pong ball, both will fall at the same rate. For you know (or at least I hope you know) that the rate of fall of any object does not depend on its weight or mass; all objects irrespective of mass fall to earth at the same rate. Therefore they will all follow the same trajectory and land at the same distance away from the passenger.

Answer 3

Ping pong ball will fall further. Since the golf ball weighs more, and both are about the same size, the golf ball will have more force per area than the ping pong ball. Therefore, the golf ball will fall faster since air resistance will have a lesser effect, therefore reaching the ground before the ping pong ball.

Answer 4

If you ignore air resistance, they will both travel exactly the same path. If you include air resistance, the ping pong will travel further from the passenger because it is in the air longer.

Although both the ping pong ball and the golf ball experience the same air resistance, the weight of the ping pong ball is much less, so the force resultant from the air resistance is significant in comparison to its weight.