Why I like the infamous Cricket Ball Scene
The episode "Four to Doomsday" is primarily known for two things: Tegan knocking out Adric for being a twerp and then victory strutting her way off the spaceship, and the Doctor messing about with Cricket Balls in space. These moments are not unrelated: Tegan's righteous wrath and desperation has left her and the TARDIS stranded some several hundred meters to the left of where they started, and the Doctor's attempts to swim out to it have left him equally stranded (Adric being a twerp probably didn't help with that either).
I say infamous because I guess people take a lot of issue with this scene for reasons of SCIENCE. I think it's brilliant though. Just absolutely screaming genius brilliant, and I have math to back it up. Here's what's going on. The Doctor is stuck in space between the Ubankan ship and the TARDIS - all three of them in the same reference frame - so functionally stationary with respect to one another. Never mind how he got there. He's got a helmet on, so he can breathe and his eyeballs won't explode and all that good stuff, and he's already told us that he can withstand vacuum for like 5 or 6 minutes anyway. The trouble is he's in an entirely frictionless environment, and any gravitational attraction between the three relevant bodies is negligible, so he can't actually get anywhere, because there's nothing to push off of. It's a serious problem. So, being the crickety cricket Doctor that he is, he takes a cricket ball out of his pocket and throws it at the spaceship. It bounces back, he catches it, and the momentum carries him easily to the TARDIS. Hooray!
I love this for a number of reasons. Firstly, it touches the same little corner of my soul that loves MacGyver - low-tech solutions to high-tech problems. Secondly, I love any scene that involves random stuff coming out of the Doctor's pockets. Thirdly, of course the cricket-themed Doctor has a cricket-themed solution to his problems! Of course he does! And fourthly, I really really appreciate it when Doctor Who relies on actual (and really very simple) science, instead of technobabble or Sufficiently Advanced Technology. Every so often the show manages to return to its educational roots, and it's just lovely. Because the thing is, this would totally work, albeit not quite as shown on screen.
This scenario is a textbook Conservation of Momentum problem. As in literally textbook. If you open an intro physics textbook to the momentum chapter, you will probably find a problem very much like this:
"You are standing motionless in the very middle of a frozen lake. Assuming an entirely frictionless surface, how do you reach the shore?"
The answer is: you throw your shoe. Momentum is a vector - it has direction. Since you have no momentum before you throw your shoe, you + the shoe must still have a total of no momentum after you throw it. Since your shoe has momentum in the direction you threw it, you must therefore have equal momentum in the opposite direction - and you slide easily to the edge of the lake and escape, sadly without your shoe (which should eventually have reached the other side of the lake, so like...just go and get it).
So what that means is: the only flaw in the Doctor's cricket ball plan is that it should have worked better than what we saw on screen. He should have started moving immediately when he threw it. And who knows - maybe he did and the camera angles weren't quite up to snuff. At any rate, I am more than willing to class it as good enough science, and I honestly don't get why folks deride it so. Any problems with it are rough edges that a show of this kind can be forgiven for sanding off - the basic concept is absolutely sound.
But I promised some math, so let's go.
momentum (p) = mv (mass times velocity)
mass of an average cricket ball (m) = .160 kg
estimated mass of the Doctor + space helmet (M) = 80 kg (based on Peter Davison's height of 184 cm, and assuming Time Lords have equivalent density to the actors portraying them)
average velocity of a cricket bowl (v) = 45 m/s (based on the Doctor showing himself to be a top-of-the-line bowler a few episodes later in "Black Orchid," and assuming he can still bowl equally well with absolutely no leverage)
And now, assuming that the TARDIS is 100 m from the Ubankan ship, and that the Doctor made it 2/3 of the way there (66 m), we should be able to solve for some things.
0 = mv + MV (conservation of momentum)
V = -(mv)/M = -.160*45/80 = -.090 m/s ~ about a centimeter per second towards the TARDIS
So, at the moment the Doctor throws the ball, he starts moving backwards slightly less than a centimeter per second. I could see the camera missing that, personally.
Now, there's nothing to slow the ball down, so assuming a perfectly elastic collision with the (immovable) spaceship, it should bounce back at exactly the same speed it went out with. So, if V2 is the Doctor's speed after he catches it, we should have:
MV - mv = (m+M) V2
MV = -mv though (from above)
V2 = 2(MV)/(m+M)
but since the mass of the cricket ball is negligible compared to the mass of the Doctor, V2 ~ 2V, or .18 m/s towards the TARDIS.
Assuming the total distance from the ship to the TARDIS is 100 m and the Doctor got about 2/3 of the way there (66 m) before stopping (somehow), and taking into account that he will already have moved a bit before he catches the ball, at this rate, he reaches the TARDIS about 190 seconds after throwing the cricket ball.
So that's just over 3 minutes in space, and while the scene doesn't take nearly that long, it's definitely survivable between his Bizarre Alien Biology and the fact that he thought to put a space helmet on. Again, firmly in the realm of "good enough" science.
The other minor quibble is that, since he catches the cricket-ball slightly off his center of mass, he should have started to spin slightly, but again, I can forgive them for eliding that admittedly delightful detail.
In conclusion: would this work in real life? YES. Is it "bad science?" NO. Is it perfect? of course not. Is it nevertheless better than most science you see in a show like this? ABSOLUTELY. Would the Doctor and MacGyver get along? Of course not, they're way too similar. Is this my very favorite scene in this whole silly episode? YOU BET. How great is conservation of momentum? THE GREATEST.
No but for serious on that last. Conservation of energy is pretty cool and all, but the mass-energy conversion can play a little havoc with that. So okay, we've got conservation of mass-energy. And then you need to worry about relativity and curved spaces but MY GUYS. CONSERVATION OF MOMENTUM. Is the best conservation law of all time. Conservation of angular momentum can come too. Look, what other conservation laws have whole sports and games and things based off of them? Not conservation of energy that's for darn sure. I am always always down for creative exploitation of conservation of momentum. Whether it involves cricket or not.
I say infamous because I guess people take a lot of issue with this scene for reasons of SCIENCE. I think it's brilliant though. Just absolutely screaming genius brilliant, and I have math to back it up. Here's what's going on. The Doctor is stuck in space between the Ubankan ship and the TARDIS - all three of them in the same reference frame - so functionally stationary with respect to one another. Never mind how he got there. He's got a helmet on, so he can breathe and his eyeballs won't explode and all that good stuff, and he's already told us that he can withstand vacuum for like 5 or 6 minutes anyway. The trouble is he's in an entirely frictionless environment, and any gravitational attraction between the three relevant bodies is negligible, so he can't actually get anywhere, because there's nothing to push off of. It's a serious problem. So, being the crickety cricket Doctor that he is, he takes a cricket ball out of his pocket and throws it at the spaceship. It bounces back, he catches it, and the momentum carries him easily to the TARDIS. Hooray!
I love this for a number of reasons. Firstly, it touches the same little corner of my soul that loves MacGyver - low-tech solutions to high-tech problems. Secondly, I love any scene that involves random stuff coming out of the Doctor's pockets. Thirdly, of course the cricket-themed Doctor has a cricket-themed solution to his problems! Of course he does! And fourthly, I really really appreciate it when Doctor Who relies on actual (and really very simple) science, instead of technobabble or Sufficiently Advanced Technology. Every so often the show manages to return to its educational roots, and it's just lovely. Because the thing is, this would totally work, albeit not quite as shown on screen.
This scenario is a textbook Conservation of Momentum problem. As in literally textbook. If you open an intro physics textbook to the momentum chapter, you will probably find a problem very much like this:
"You are standing motionless in the very middle of a frozen lake. Assuming an entirely frictionless surface, how do you reach the shore?"
The answer is: you throw your shoe. Momentum is a vector - it has direction. Since you have no momentum before you throw your shoe, you + the shoe must still have a total of no momentum after you throw it. Since your shoe has momentum in the direction you threw it, you must therefore have equal momentum in the opposite direction - and you slide easily to the edge of the lake and escape, sadly without your shoe (which should eventually have reached the other side of the lake, so like...just go and get it).
So what that means is: the only flaw in the Doctor's cricket ball plan is that it should have worked better than what we saw on screen. He should have started moving immediately when he threw it. And who knows - maybe he did and the camera angles weren't quite up to snuff. At any rate, I am more than willing to class it as good enough science, and I honestly don't get why folks deride it so. Any problems with it are rough edges that a show of this kind can be forgiven for sanding off - the basic concept is absolutely sound.
But I promised some math, so let's go.
momentum (p) = mv (mass times velocity)
mass of an average cricket ball (m) = .160 kg
estimated mass of the Doctor + space helmet (M) = 80 kg (based on Peter Davison's height of 184 cm, and assuming Time Lords have equivalent density to the actors portraying them)
average velocity of a cricket bowl (v) = 45 m/s (based on the Doctor showing himself to be a top-of-the-line bowler a few episodes later in "Black Orchid," and assuming he can still bowl equally well with absolutely no leverage)
And now, assuming that the TARDIS is 100 m from the Ubankan ship, and that the Doctor made it 2/3 of the way there (66 m), we should be able to solve for some things.
0 = mv + MV (conservation of momentum)
V = -(mv)/M = -.160*45/80 = -.090 m/s ~ about a centimeter per second towards the TARDIS
So, at the moment the Doctor throws the ball, he starts moving backwards slightly less than a centimeter per second. I could see the camera missing that, personally.
Now, there's nothing to slow the ball down, so assuming a perfectly elastic collision with the (immovable) spaceship, it should bounce back at exactly the same speed it went out with. So, if V2 is the Doctor's speed after he catches it, we should have:
MV - mv = (m+M) V2
MV = -mv though (from above)
V2 = 2(MV)/(m+M)
but since the mass of the cricket ball is negligible compared to the mass of the Doctor, V2 ~ 2V, or .18 m/s towards the TARDIS.
Assuming the total distance from the ship to the TARDIS is 100 m and the Doctor got about 2/3 of the way there (66 m) before stopping (somehow), and taking into account that he will already have moved a bit before he catches the ball, at this rate, he reaches the TARDIS about 190 seconds after throwing the cricket ball.
So that's just over 3 minutes in space, and while the scene doesn't take nearly that long, it's definitely survivable between his Bizarre Alien Biology and the fact that he thought to put a space helmet on. Again, firmly in the realm of "good enough" science.
The other minor quibble is that, since he catches the cricket-ball slightly off his center of mass, he should have started to spin slightly, but again, I can forgive them for eliding that admittedly delightful detail.
In conclusion: would this work in real life? YES. Is it "bad science?" NO. Is it perfect? of course not. Is it nevertheless better than most science you see in a show like this? ABSOLUTELY. Would the Doctor and MacGyver get along? Of course not, they're way too similar. Is this my very favorite scene in this whole silly episode? YOU BET. How great is conservation of momentum? THE GREATEST.
No but for serious on that last. Conservation of energy is pretty cool and all, but the mass-energy conversion can play a little havoc with that. So okay, we've got conservation of mass-energy. And then you need to worry about relativity and curved spaces but MY GUYS. CONSERVATION OF MOMENTUM. Is the best conservation law of all time. Conservation of angular momentum can come too. Look, what other conservation laws have whole sports and games and things based off of them? Not conservation of energy that's for darn sure. I am always always down for creative exploitation of conservation of momentum. Whether it involves cricket or not.