# Physicists.. can you help settle a dispute.

I present a scenario to you, and I will also present two possible outcomes. Can you please inform me as to which is the correct outcome.

You are standing on top of a train which is travelling east at the speed of a bullet. You are positioned at the west end of train, facing west. You raise a pistol and aim it west, and fire it. What will the bullet do?

A) More or less, drop straight down (not travel in east or west)

or

B) Travel west at (more or less) the speed of a bullet

I am saying it’s A, my mate is saying B. Please help settle once and for all.

I would have to go with A. The bullet should move at the speed of a bullet, reletive to the train.

Allowing for air resistance however, the fact that the bullet would have a “tailwind” would mean that it would go westward at least somewhat.

A - The bullet will hit the ground directly under the spot where it was fired. The velocities will cancel. From the person firing the gun the bullet will move away at the speed of the bullet. To an observer on the ground the bullet drops straight down.

This is all different if you talk about the speed of light, but were talking Newtonian physics here.

Neither are correct. It will travel at twice the speed of a bullet with respect to a stationary observer.

You’re firing a gun backwards from the rear window of the train, right? And by “speed of a bullet,” you mean muzzle speed? If you neglect air resistance, the bullet should drop straight down. If you don’t neglect air resistance, the air behind the train is being dragged by the train, so I think the bullet would move slightly backwards (i.e. to the east) before it hits the ground.

“B” is more correct but not totally. Let’s say the train is moving east to west at 2000 feet per second (a good average velocity for a rifle bullet). You fire a rifle in the direction that the train is going in and it leaves the barrel at an additional 2000 fps. That means that it has an initial speed of 4000 fps to an observer on the ground and 2000 fps to the shooter. However, things are not normal here for the bullet. The headwind is absolutely horrible so it will rather rapidly slow down and not travel nearly as far as if it was shot from the ground. The bullet will travel some distance relative to the shooter but performance will be greatly degraded over a bullet shot into such a strong headwind.

Oh sorry. I think that I got the direction that you are firing the gun wrong. Please disregard my answer above because it is not correct as your question is worded.

AH! The age-old funnily-worded question problem.
Train travels west at 2000 fps.
Bullet shot from moving train, going 2000 fps east.

The bullet, if there’s no atmospheric friction (we’ll ignore that), will seem to drop straight to the ground, as seen by a stationary observer.

Now we get to the fun stuff - if we assume that air is at zero fps, stationary, then it will have no effect on the bullet, and the bullet will simply drop.
BUT THIS IS NOT HOW AIR AROUND A MOVING TRAIN IS! The air is moving in a very complex way, which I do not have enough physics knowledge to describe correctly.

Suffice it to say that the bullet will land soemwhere NEAR where it is fired.

Actually, it would behave in similar way to a bullet dropped into a 2000 ft/sec headwind (ignoring the vortex effect of the shooter). Thus, all we need is a good windtunnel to perform this experiment and find out how the bullet would travel. Lacking a wind tunnel, we can approximate it by assuming the bullet has a 1cm^2 surface area exposed to the wind at all time, you could work out roughly where it would land. Im not a mech engineer so you would have to wait for one of those to pop in.

I don’t think so. The air is stationaryrelative to the ground, plus the turbulence. Initially, the bullet is also stationary relative to the ground. So it’s like tossing a bullet into the wake of a train.

Of course, no land-based vehicle can travel faster than a bullet so the question is academic anyway.

You could say that both the answers are correct, because the answers have no frame of reference. If you were riding on the train, and had sharp enough eyes to track a bullet, you would see the bullet recede into the distance just like you’d fired it from a stationary platform. On the other hand, if you were sitting on the ground watching (or attempting to watch) the train go by, you would see the barrel of the gun apparently deposit an unmoving bullet, which would promptly be buffetted around by the train’s wake. A train travelling at supersonic speeds (BTW, there have been a few rocket cars that exceeded the speed of sound on the ground - or this could be a subsonic bullet) is going to leave quite a wake, but there’d probably (it’s very hard to predict!) be a vaccuum off the tail of the train that would pull the bullet along behind the train briefly before the bullet dropped to the ground.

Now, if we substituted a slower train and something that moved slowly (you could probably pull this off with an arrow, for instance) the train would have a much smaller wake, the air effects would drop out of the picture, and if you were watching from the ground the projectile would simply fall straight to the ground.

Wouldn’t an Amtrak ticket be cheaper?

The Thrust SSC hit 766.109 MPH, 1123.63 fps. Many bullets travel slower than that, some a lot slower.

Definitely A (again ignoring the wake of wind drawn along by the train)

Thanks for the correction. I thought almost all bullets were supersonic.

I am not a physicist, but I once had an insane high school physics teacher. If you fire a bullet perpendicular to the ground, it will be pulled to earth with an acceleration of g, just like any other falling object, which will give it a parabolic path. It will hit the ground in a fraction of a second, assuming you are only a few feet off the ground. You can calculate the distance it will travel by knowing the normal muzzel velocity for your round and weapon, and how far it can go horizontally during the height of its fall at the rate of 32 feet per second squared. Air resistance will slow it down slightly, so adjust for the coefficient of friction, mu.

Now, you have added that the train is travelling in the opposite direction of the bullet being fired, at the muzzel velocity of the bullet. There will be quite a wake, or draft messing up air velocity at the rear of the train from whence you are firing, and that will mess up mu quite a bit, but I gather that mu isn’t really part of the problem. Neither is the amount of energy the recoil of the gun adds to the velocity of the train, which is negligible.

I’ve got to go with A the bullet hits the ground near the point of the earth a few feet lower than when it was originally fired. To suggest B, that the bullet travels from the point of firing, ignoring the negative velocity of the train, and basically giving the bullet a much larger gunpowder charge to go twice as fast. The exception I can think that would make B true would be if you were not at the end of the train and firing inside the train, it would look like you weren’t on a train at all, but still hit the floor of the compartment approximately above the point it was fired.

However, the bullet will not drop to the ground, either. Consider the situation where the bullet is fired one meter from the end of the train. Nothing else has changed.

With reference to the train or people on it, the bullet moves at a speed v regardless whether it has been fired at the end of the train or one meter from it. If it drops to the ground, that means the bullet suddenly stops.

Now remember that the train moves in the opposite direction of the bullet. If there’s any wind, it will be tail wind.

One thing that confuses people more than anything else (I think) when considering questions like this is that (disregarding the curvature of the earth and spin and atmosperic effects), the lateral velocity of an object does not affect the rate at which it accelerates toward the earth under the influence of gravity.

In a vacuum, on a perfectly flat surface, a bullet fired from a rifle will hit the ground at exactly the same time as one simply dropped from the same height as the fired bullet exits the barrel; that the fired bullet has a very large horizontal velocity component does not in any way affect it’s descent to the ground under the force of gravity.

None of that is directly relevant to the question, I know, but I do feel that it’s easy to start thinking of bullets as special cases.

Anyway, to the OP; lets simplify the whole thing:
-Suppose you have a machine that will launch a projectile at exactly 100km/h
-You put it on the back of a train, aiming it exactly opposite to the direction of travel
-You put on your space suit and the train enters a very long tunnel from which all the air has been evacuated.
-The train reaches 100km/h and you activate your machine, launching the projectile.
-From your point of view, the projectile recedes from view at exactly 100km/h, as designed.
-From the point of view of a (space-suited) stationary observer, a train passed by at 100km/h and dropped a projectile out of the back, which fell vertically to the ground.

It’s the same thing - it’s the idea of bullets falling straight to the ground that is uncomfortable to grasp, but as I said above, bullets do fall straight to the ground, they just do it with a big sideways velocity component.

If you want the stationary observer to see the projectile racing away from the train at 100km/h, then you have to launch it at 200km/h from the train.

To simplify it still further, let’s call it a “gun-like device”?