Built on Facts

There’s a question that gets posed toward the beginning of intro physics classes to gauge the students’ understanding of acceleration. If you fire a bullet horizontally while at the same instant dropping a bullet from the same height, which hits the ground first?

The point is to think clearly about the equation describing accelerated motion. The equation is this:

The bold letters represent vectors. Lower case r is position, a is acceleration, v is velocity. The vectors with 0 subscripted represent the starting values, so v0 is the initial velocity and r0 is the initial position.

But the whole point of a vector equation is that the equation compactly describes the motion in 3-d space. If we pick coordinates such that the x-axis is parallel with the ground and the y-axis is vertical, the equations remains true if we replace r with x (or y), a with the acceleration in that x or y direction, and v0 with the initial velocity in that direction.

So consider that fired bullet and that dropped bullet. Both of them have the same acceleration in the y direction (-9.8 m/s^2, vertically downward, from gravity). Both have the same initial velocity in the y-direction (0, since the fired bullet’s initial velocity is all in the x direction). And both have the same initial position in the y-direction – call it h, the initial height. That means both bullets are obeying the same equation in the y-direction, so they both have to reach the ground at y = 0 at the same time.

When we replace r with x, the x-direction equations are different for the two bullets, since the dropped bullet has an initial velocity of 0 in the x-direction while the fired bullet has a large initial velocity from being fired. But that matters not at all to the vertical motion.

Incidentally we can use this to find the range of a horizontally fired bullet. Take the y-direction equation, set y = 0 (which is when the bullet hits the ground), solve it for t, and plug that t back into the x-direction. The x-distance attained during that time is just the range. Doing this, I get:

For a bullet at 1,000 feet per second fired at 5 feet, that’s about 550 feet. (304 m/s and 168 m)

But until relatively recently, no one had ever actually done the experiment. It’s difficult, both in terms of dropping the bullets properly and making sure the gun fires exactly horizontally. Horizontal fire is critical, because if there is an initial vertical velocity for the fired bullet, the equation will be different from the dropped bullet and they won’t hit the ground at the same time. Nonetheless there is a group of experimenters who are very good at this sort of thing, and not so long ago they actually did the experiment. They are of course the Mythbusters:

They do the experiment and the bullets land within a few milliseconds of each other. Of course things aren’t quite mathematically perfect in real life, as air resistance and other factors will serve to distort the basic acceleration equation we’re using. Nonetheless it’s a quite good approximation and in this case works out beautifully.

Comments

#1 D. C. Sessions
October 19, 2009

An utterly lovely extra-credit assignment is to figure out how long it takes for the nonlinearity of air resistance to cause a given discrepancy.

Don’t try to derive a closed-form solution unless you’re into pain.
#2 Andrew Foland
October 19, 2009

At the millisecond level, the correct answer to this question depends on whether the planet you live on is shaped like a plane or a sphere.
#3 _Arthur
October 19, 2009

In the Mythbusters episiode, I’m not statisfied they double-checked it the gun and the dripped bullet were precisely at the same height.

They had a laser pointer, but I didn’t see them use it for that purpose.
I think they assumed (they stated it) that the floor was perfectly level.

In any case, despite the possible half-a-millimeter height uncertainty, and the problems with accurately dropping the bullet, the experiment was a success.
#4 Geoff
October 19, 2009

If I were teaching a physics class right now, I would ask the class if this is a convincing result, or how much room it leaves for a) doubting the core phenomenon (motion in y independent from motion in x), and b) the room left for considering other phenomenon (aerodynamic effects, for example). The time difference on impact was 36 ms. For a fall of one meter, the expected drop time is ~450 ms. So, this is a difference on the order of 10%.
#5 NoAstronomer
October 19, 2009

We actually did this experiment, slightly modified, in my high school physics class. The teacher used a marble and a spring (think pinball) for the bullet and the gun.

For the target we used a small can. The can was held in place by an electro-magnetic. When the current to the magnet was switched off the spring was released at the same time. The target and the gun were about 6 feet apart, both about 3 feet off the floor.

The marble never touched the floor – it was inside the can.
#6 Robert
October 19, 2009

They should have had a grenade launcher fired horizontally compared to a dropped grenade falling straight down.

No myth is confirmed until Adam throws up and there is a big explosion at the end.
#7 CCPhysicist
October 19, 2009

@5:

The tolerance in your experiment is really large unless the can has the same diameter as the ball.
#8 Mark Eichenlaub
October 19, 2009

D. C.

I worked a little on counting in air resistance. I think it should be right on the fringe of being a contributing factor to the uncertainty in the experiment even as it stands.

calculation here
#9 mrcreosote
October 19, 2009

“Horizontal fire is critical, because if there is an initial vertical velocity for the fired bullet, the equation will be different from the dropped bullet and they won’t hit the ground at the same time.”

During my (Australian) Army Reserve training, when firing an SLR rifle we were told that for short ranges (under ~100m) you needed to aim low because the bullet actually went up as it left the barrel.
#10 Robert
October 19, 2009

If you did the experiment under water in a swimming pool, the fired bullet would quickly lose its forward speed but gravity would keep pulling it to the bottom, and gravity would have the same effect on a dropped bullet in water.

So why should the resistance of the medium make a big difference if both bullets have the same shape and density and both are falling with the same gravitational force in the same medium?

The MythBusters ensured that the flying bullet did not have enough positive or negative lift to raise its height measurably or make it fall faster. That’s all you need to show.
#11 Mark Eichenlaub
October 19, 2009

Robert,

The point is that air resistance doesn’t make a big difference, but it makes a difference. This is because drag is not linear in velocity.

The lift on the bullet is another issue entirely.
#12 magista
October 19, 2009

This video is going up on my website for my students tomorrow!
#13 Tex
October 19, 2009

During my (Australian) Army Reserve training, when firing an SLR rifle we were told that for short ranges (under ~100m) you needed to aim low because the bullet actually went up as it left the barrel.

IANAP, but I suspect that this is just plain wrong. I think it is more likely that the sights were adjusted to some moderate distance (maybe 500m?), and that for any shorter distance the sighting would over-compensate for the drop due to gravity.
#14 Robert
October 20, 2009

That video is no good because whoever stole it cropped the bottom off to get rid of the Discovery logo and then they covered it with a link to their web site that features stolen videos. Because of the cropping you can’t even see the shot bullet hitting the floor when the falling one hits. Didn’t anybody notice this???

I think you need to remove the video and link to one that is not stolen, or at least one that isn’t cropped.

Thank you.
#15 Matt Springer
October 20, 2009

The “bullet goes up as it leaves the rifle” is definitely not right. It’s one of the very numerous bullet myths out there. It’s right up there with “.50 BMG can kill with a near miss”. In fact, Mythbsters did that one too, come to think of it.

As for the video link, I presume Discovery isn’t too perturbed about it or they’d send Youtube a takedown notice for that and the numerous other Mythbusters videos. If that happens, I suggest watching the episode when they replay it on Discovery.
#16 Robert
October 20, 2009

Let me explain it again, Einstein.

The video you posted is cropped, which means the bottom of the screen frame is cut off.

Are you with me so far, Einstein? Hang in there with me.

The high speed video that the MythBusters showed of the bullets hitting the floor had the camera set up so that the bullets landed near the very bottom of the frame.

So you see, Einstein, your video shows nothing of the bullets landing on the floor except the normal speed shot which is too fast for the human eye to see.

You posted a SPAM video that teachers will show to children, AND IT IS SPAM, and it does NOT even show the children the physics that you wanted them to see because the part you wanted them to see is cropped out, IT ONLY SHOWS THEM SPAM.

It’s not whether Discovery is perturbed. It’s about you not giving a rat’s ass about the crap you posted, Einstein.
#17 Greg Laden
October 20, 2009

It is totally disappointing that they kludged their methods.

Your equation is wrong. It assumes that the earth is flat. They should be able to use the difference between the dropped and fired bullet to calculate the size of a sphere roughly representing the earth. (Assuming that when the floor was laid it was leveled with preternatural precision).

Back to the drawing board!!!
#18 Matt S
October 20, 2009

For another interesting analysis on this same Mythbusters show http://www.dotphys.net did this a few weeks ago. Cheers.
#19 Brian H
October 20, 2009

Wow, Robert really has his panties in a twist. Is he like this often Matt?
#20 Anonymous
October 20, 2009

“During my (Australian) Army Reserve training, when firing an SLR rifle we were told that for short ranges (under ~100m) you needed to aim low because the bullet actually went up as it left the barrel.”

I believe this is because a line drawn between the sights and a line drawn parallel to the barrel are not parallel to each-other. So the bullet “goes up” relative to the plane the sights are in. The lines formed by the sights and the parabolic path of the bullet (ideally) intersect as the bullet is falling from above the sight plane at the distance the weapon is sighted at.

I assume that the mythbusters leveled relative to the barrel, so this is not a concern.
#21 Brian H
October 20, 2009

oops, #20 was supposed to be me too. Sorry Matt
#22 ppnl
October 20, 2009

Years ago I worked putting together physics demos at TAMU. One we did was called the monkey shoot. Basically we used compressed air to blow an aluminum slug toward a can suspended by an electromagnet. When the slug leaves the tube it hits a micro switch which released the can. Since they fall together the slug must hit the can.

The thing is the gun and the can do not have to be at the same level. As long as the gun is pointed at the can it can fire up at 45 degrees and still hit it.

Well it didn’t work very well. Usually the microswitch misfired and even when it didn’t it threw the timeing off. We ended up replacing it with photocells and were getting accuracy for as far as the gun would fire. It would plonk the can just inches before they hit the floor.
#23 rob
October 20, 2009

@Greg Laden:

they would have gotten away with it, if it weren’t for meddlin’ biological anthropologists.

heh.
#24 Skwid
October 20, 2009

A lot of the criticisms of the posters here are answered by Adam and Jamie in this “Combustion Chamber” video, which is apparently something they’re going to do semi-regularly now?
#25 Max
October 21, 2009

In The Mechanical Universe the lecturer at the end does a variant of this experiment at one point where he fires a dart gun at a falling target. Not the same precision or accuracy, but a fun exhibition.
#26 Jim Thomerson
October 21, 2009

Back in the ’50′s battle sight for the M1 was 300 yards. So the bullet would rise above the line of sight and drop back to the line of sight at 300 yards. I think the bullet passes through the line of sight on the way up at 100 yards, but don’t recall for sure. Anyway, you sight the same to get bulls at 100 and 300 yards.

Curvature of the earth is 0.6 ft/mile. In surveying, you add on a little because of the atmospheric refraction resulting from the earths curvature. My surveying book is out in the shop, but it is maybe an additional 0.3 feet. I suspect your house was built on a flat earth.