Built on Facts

You know what irritates me? NCIS. Not the Navy’s law enforcement agency, the TV show about that organization. I’m generally a fan of Law and Order style police procedurals, but that one just rubs me the wrong way. Why? The fetishize law breaking. Not the law breaking of the criminals they investigate, they themselves go out of their way to break the law. Need a DMV record? Hack their computers, don’t bother to actually call and ask them or get a warrant. Suspect not talking? Give the Mossad officer on loan to NCIS fifteen minutes alone with the suspect. A relative accused of murder based on overwhelming evidence? Hide her from the police and obstruct the investigation. Anyway, it rather defeats the whole “rule of law” thing democratic nations like to have and they do it every episode, usually repeatedly.

“But Matt,” you implausibly say, “you like 24 and it’s much worse by that standard.” Yep, that’s true. But 24 is deliberately preposterous – like criticizing Star Trek because all the aliens look like humans, dissecting civil rights problems in 24 misses the point. But shows like NCIS are explicitly about real law enforcement agencies acting in typical circumstances. I’m therefore not happy about their tendency to completely ignore the law in the name of enforcing it.

Is there going to be any physics in this post? Sure. Totally unrelated to any of the above, but hey, at least the laws of physics aren’t breakable. In fact, let’s pose this one as a question. Purely qualitative, I’m not going to make you solve the equations.

A ball with some initial velocity v rolls without slipping up a hill of height h. The top of the hill is actually the edge of a cliff, so when the ball reaches that height h it flies horizontally off the cliff and falls back to the ground that same height h below. There is no rolling resistance or air resistance. The ball hits the ground with a total velocity:

a) less than v
b) equal to v
c) greater than v

Why?

This (with numbers) was a homework problem my 201 students had to do. It’s one of the more interesting problems in the chapter. I’d have given it to them as a quiz problem but the algebra is a little tedious for the ten minutes or so they have to take the quiz. Conceptually though that multiple choice question can be answered without any math, just a little thought.

I’m sure someone in the comments will get it right, but if not I’ll post the answer tomorrow.

Comments

#1 PPNL
October 22, 2008

Well when it falls to the ground it isn’t moving anymore so I pick a.

Not what you intended? Ok without friction the ball cannot be moving at a lesser velocity and without more Ke input it cannot be moving faster. Gravity only transforms energy between Ke and Pe. So b.
#2 Eric Lund
October 22, 2008

I see PPNL got it first, but it’s simple. There are no dissipative forces here, and the potential energy at start and end are the same by construction. Therefore the kinetic energy must be the same, which means that the speed* must be the same, and (b) is the correct answer.

*I assume that this is what you mean by “total velocity”. The actual vectors will be different because the hill exerts a normal force with a horizontal component, and there is no horizontal force after the ball rolls over the cliff.
#3 Karl Withakay
October 22, 2008

It’s been a few years since I took engineering physics in college but,

The total energy of the system must be conserved. No energy is lost through friction or drag, and the potential energy is the same in the initial and final states, therefore the initial and final kinetic energy are the same. The mass has not changed, so Vf=Vi, assuming we’re really concerned with the scalar of speed rather than the vector of velocity.
#4 Karl withakay
October 22, 2008

PS about NCIS:

I don’t delude myself about NCIS being the least bit realistic; it’s just simple entertainment with some nice eye candy thrown in. (Abby vs Ziva = Mary Anne vs Ginger ?)

Besides all the criminal law enforcement activities, it’s absurdly easy for for McGee (field agent) and Abby (forensic scientist) to hack into anything they want to.
#5 Anonymous
October 22, 2008

“*I assume that this is what you mean by “total velocity”.”

Reminds me of an old trick question. A race car driver drives around a track three times at 100mph, five times at 150mph and seven times at 200mph.

Q)What was his average velocity?

A)Zero!
#6 Zifnab
October 22, 2008

In all fairness, getting on Jack Bauer’s case about unnecessary roughness is like getting on James Bond’s case about sexual harassment. They’re – for all cinematic purposes – the same unbelievable character.

Also, I’m guessing a) and trying to figure out exactly why. I’m betting it has something to do with the fact that the hill is sloped and the ball has to move a greater distance to get to the top of the hill than it does to get back down.
#7 Ned Wright
October 22, 2008

“rolls without slipping” is not irrelevant. A ball rolling down a ramp accelerates more slowly than a frictionless block sliding down the ramp. Energy is conserved but you have to include rotational kinetic energy. The ball starts with velocity v and rotational speed omega = v/radius. At the top it has lower speed and lower omega, and this lower omega is retained while falling, so the ball lands with less rotational kinetic energy that it started with and it has to have a higher v. So (c).
#8 Dan
October 22, 2008

It’s been a long time since I’ve worked this sort of problem, but I agree rotational inertia and ‘not slipping’ are the key.
#9 Andy Wood
October 22, 2008

The correct answer is (c).

The translational kinetic energy of the ball is mv^2/2.
It rolls without slipping, so its rotational kinetic energy is I(v/r)^2/2, where I is its moment of inertia and r is its radius.

As it rolls uphill, both translational and kinetic energy are transferred to potential energy. However, after it falls off the cliff, all of that potentail energy is converted back into translational kinetic energy, none into rotational kinetic energy.

Therefore, on reaching the ground, some rotational kinetic energy has been transferred into translational kinetic energy and the final speed must be greater than the initial speed.
#10 Andres Villarreal
October 22, 2008

I agree with #7:
- the initial potential energy is the same as the final
- initial (kinetic + rotational) energy is the same as the final
- initial rotational energy is higher than the final
- there are no other kinds of energy
- the final kinetic energy has to be greater than the initial, to compensate
- the kinetic energy grows with speed, so the final speed is greater than the initial.

Although the entertainment business has some degree of “creative license”, we must be angry when boundaries are crossed. The issue of morality is especially dangerous: people are really thinking that crimes in the name of “the good cause” are acceptable. Even Anthony Kennedy, the Supreme Court Justice, is saying that the good guys can torture the bad.
#11 kevin
October 22, 2008

Ned’s right. Assuming the ball was rolling at the start, and that it has mass.
#12 DG
October 22, 2008

Well, it’s already been figured out, but this is a classic case of a problem for which you get an answer in ~2 seconds, spend some time patting yourself on the back, and then realize (relatively) much later that you’ve completely missed the hangup.
#13 Chris
October 22, 2008

NCIS is one of the few procedural dramas that I can’t enjoy.

If you like comics, check out Alan Moore’s Top 10 or Brian Michael Bendis’ Powers. Both (especially the latter, with non-powered cops and a male/female pair as main characters) do a good super-interpretation of the whole crime drama mythos.
#14 Matt Springer
October 22, 2008

Ned is the first to the right answer, and wins! It’s kind of counterintuitive, but the ball will by his reasoning hit the ground with a higher speed than it started.

To clear up any confusion, by “total velocity” I just meant “speed”. I wanted to make sure we weren’t just talking about the vertical or horizontal components of the velocity individually.
#15 Matthias
October 22, 2008

I actually like NCIS because of how screwed up all these people are. Yeah, their morals are� somewhat wavering, but maybe it’s actually being realistic?
#16 CCPhysicist
October 22, 2008

I need to get over here earlier in the day. I can’t believe it took 6 wrong answers to get to a correct one, but I’m sure very few of my students would get it right.

But why algebra? It is just that the final KE is the same but the rotational KE is a smaller fraction because it remains unchanged from when the ball was at the top.

A very nice conceptual question, not unlike rolling a hoop and a sphere up a hill with the same initial speed at ground level:
a) sphere goes higher
b) same height
c) hoop goes higher
#17 CCPhysicist
October 22, 2008

I don’t quite get your problem with NCIS. Breaking the law to save it was standard operating procedure before Miranda and long after (viz. FBI during the 60s and 70s and NSA more recently). Check out “Life on Mars” for a flashback to 1973. Does it bother you because, being less hyperbolic, it strikes you as being more likely to be true?

If you read the reports, you’ll see that both shows represent what has been going on in some parts of our government, and not just in the Bush administration (just more of it). I just hope that the guilty parties don’t shred every bit of evidence before either McCain or Obama come into office, seeing as both of them have opposed torture as an interrogation method.
#18 Matt Springer
October 23, 2008

I don’t think have any illusions about what sometimes goes on in law enforcement, but I can’t see any way glorifying abuse of power for humor value is something we ought to let pass without comment. Imagine Everybody Loves Raymond if Ray beat his wife a few times an episode to the sound of a laugh track. Like law enforcement casually breaking the law, it would defeat the entire purpose of the show.
#19 ralph137
October 23, 2008

If you whack a ball up an incline and there is no rolling resistance why would it roll up? Wouldn’t it just slide up?
If you hit it with insufficient V up and it does roll, will it not come down with very close to same V.
If you hit it with extra V up it will still move in a vertical direction as well as horizontal.
If you hit it with perfect V why would it have only a horizontal component?
Perfect V would result with half the time rolling back down with same V. Half the time falling over and down with same V.
#20 Paul Murray
October 23, 2008

Greater than v.

The ball slows down as it goes up the ramp, meaning that some of the KE from it’s rotation gets converted into gravitational PE. When it falls, all that gravitational KE is converted into linear KE – the fall does not cause the ball to resume spinning.

IOW: its initial rotation and rotational inertia means that it’s travelling faster at the top of the slope than it would be doing if it was sliding. This means that it falls as if it were a sliding object that had been propelled faster than v in the first place.
#21 John
October 24, 2008

TV Cop shows, now this is my area of expertise.
#1 Police Story (based on Joseph Wambaugh’s books. head and shoulder above the rest)
#2 Andy Griffith (a lot more realistic than you may think)
#3 Adam 12 (police work like it should be, but their defensive tactics leave a lot to be desired)
#4 Dragnet (how can you not love straight laced Sgt. Friday and Officer Gannon?)
#5 Law and Order (I’ve only watched this a couple of times, but it’s the only newer cop show that I’ve been able to stand watching for more than a few minutes)
#22 IBY
October 27, 2008

Yay, 24! I can’t wait for the next season.
#23 Jennifer Haley
August 17, 2009

they were going to the pocilce area .
#24 Lach
January 6, 2010

The Velocity is none of those, it’s different. Velocity involves direction so you can’t have increased or decreased velocity, only speed. As for speed, it is an impossible situation. As there is no rolling or air resistance, it must be in a vacuum in zero gravity. But if it is in zero gravity, there is no way for it to fall, it would just continue as it is.
#25 ball valves
March 27, 2011

The magnonomous truth about this blog article is, that it spoke to me deeply. Thank you for sharing your thoughts and concerns.