# Will a plane on a treadmill take off? **URL:** \ **Category:** Factual Questions\ **Created:** [December 7, 2005, 9:19pm UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567 "2005-12-07T21:19:53Z")\ **Posts on this page:** 20\ **Page:** 5
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [December 8, 2005, 5:00am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/81 "2005-12-08T05:00:21Z")
> [@treis](#): > > …First off let me say that it is impossible for a tire to rotate without an applied force at some point. > > …Thats false, if the treadmill starts up with no other applied force on the wheel the wheel will not rotate and merely move along with the treadmill. Wheel bearing friction has nothing to do with this situation is you assume the wheel can rotate without slipping. All bearing friction will do is cause loss of energy. > > …In both situations the plane must apply some thrust to remain stationary as the treadmill starts up. If there are no forces then the wheel would not spin. Since it is spinning we know there is a friction force point backwards. For the plane to remain stationary there must be a force opposing that… :smack: I can’t believe I made that mistake! You’re absolutely right, of course. > [@treis](#): > > Saying so is flat out wrong and someone saying so that has taught college physics is a bit frightening… Oh well, it has been 4 years since I last taught. I’m either getting rusty or stupid in my old age. ---
**Author:** ![The\_Controvert](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@The\_Controvert](https://boards.straightdope.com/u/The_Controvert)\ **Post date:** [December 8, 2005, 5:07am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/82 "2005-12-08T05:07:15Z")
The OP is correct, the plane would not lift off, because there is insufficient wind force to generate lift if the plane is not moving forward. The constraints of the problem state the treadmill is moving in the opposite direction of the wheels at the same speed as the wheels. Therefore, the plane cannot be moving forward regardless of thrust, friction, or any of the other previously raised points. The originally posed problem does not mention anything about assuming the wheel bearings are frictionless. Instead of imagining a full sized 747, think of a balsa rubber-band propeller model airplane with really strong wheels. ---
**Author:** ![Manduck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manduck/32/256_2.png) [@Manduck](https://boards.straightdope.com/u/Manduck)\ **Post date:** [December 8, 2005, 5:15am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/83 "2005-12-08T05:15:31Z")
Let’s look at it this way: The plane’s engine should accelerate it down the runway regardless of what the treadmill does. BUT we are given that the treadmill’s speed exactly matches that of the wheels, so the plane CAN’T be making any progress down the runway (because that would involve the wheels moving faster than the treadmill). THEREFORE the plane’s engine must not be working. Maybe it’s out of gas or something. ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 8, 2005, 5:19am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/84 "2005-12-08T05:19:18Z")
> [@Giraffe](#): > > Let me rephrase: we’ve got a guy with rollerblades and a jet pack on top of a train. He starts at one end, turns it on and accelerates down the length of the train. For the purpose of this example, there is no air resistance. I am asserting that there exists some possible function of acceleration vs. time that can be applied to the train which will keep the rocketman motionless with respect to the surrounding environment. This is a clear and definite statement. It’s also an erroneous one. The train can keep the rocketman motionless only if it can exert sufficient force on him. If it can exert not force, it can have no influence on his motion with respect to any other frame of reference (e.g. the earth). It can certainly influence his apparent motion with respect to itself, but that’s obvious and of no significance. If the train can’t exert force on rocketman, then there’s also no significance to the fact that he happens to be quite close to it. The guy rollerblading in the park three miles from the train tracks is equally influenced by the train’s motion, which is to say: not in the slightest. > [@](#): > > how is that different from the airplane problem? Your implication is quite correct - the two problems are the same. Depending on assumptions about friction, the treadmill either has a small effect on the plane, or none. In neither case can it prevent the plane from accelerating to flying speed. ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [December 8, 2005, 5:23am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/85 "2005-12-08T05:23:11Z")
> [@Xema](#): > > This is a clear and definite statement. It’s also an erroneous one. > > The train can keep the rocketman motionless only if it can exert sufficient force on him. If it can exert not force, it can have no influence on his motion with respect to any other frame of reference (e.g. the earth). It can certainly influence his apparent motion with respect to itself, but that’s obvious and of no significance. > > If the train can’t exert force on rocketman, then there’s also no significance to the fact that he happens to be quite close to it. The guy rollerblading in the park three miles from the train tracks is equally influenced by the train’s motion, which is to say: not in the slightest. > > Your implication is quite correct - the two problems are the same. Depending on assumptions about friction, the treadmill either has a small effect on the plane, or none. In neither case can it prevent the plane from accelerating to flying speed. This wrong, wrong, wrong. Stop posting erronous information. **Giraffe** you can indeed do what you are asking and I will post how in a few minutes but I gotta go put laundry in the dryer. ---
**Author:** ![borschevsky](https://avatars.discourse-cdn.com/v4/letter/b/97f17d/32.png) [@borschevsky](https://boards.straightdope.com/u/borschevsky)\ **Post date:** [December 8, 2005, 5:36am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/86 "2005-12-08T05:36:13Z")
> [@robby](#): > > If there is a net force acting on the plane due to its engines it will accelerate with respect to the ground. During this _accelerated_ motion, the plane’s wheels will always be turning faster with each passing second. The treadmill will constantly be trying to catch up. This leads to a loop, I think, where the wheels and treadmill speed up infinitely. I agree with this, depending on what is meant by “speed of the wheels”. Say the plane does take off; when it does, it has some positive speed X relative to the ground. The treadmill is going in the other direction at some speed Y, so the wheels are turning at X+Y. But the OP said that the treadmill always goes at the same speed as the wheels, so this is impossible. That’s unless you allow Y to be infinite, so that X+Y = Y. ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 8, 2005, 5:37am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/87 "2005-12-08T05:37:53Z")
> [@The Controvert](#): > > … the treadmill is moving in the opposite direction of the wheels at the same speed as the wheels. Okay, let’s pick a sample case: No wind. Piper Cub (which flies happily at 45mph), is rolling north at 50mph, along with its wheels. Treadmill is thus moving south at 50mph. > [@](#): > > Therefore, the plane cannot be moving forward regardless of thrust, friction, or any of the other previously raised points. Why not? The wheels are spinning at a rather brisk clip (equivalent to rolling at 100 mph) but that’s no big deal. The example above meets all your specifications and has the plane moving forward at comfortable flying speed. > [@](#): > > The originally posed problem does not mention anything about assuming the wheel bearings are frictionless. Correct - and there’s no need to. ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 8, 2005, 5:42am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/88 "2005-12-08T05:42:21Z")
> [@treis](#): > > This wrong, wrong, wrong. Stop posting erronous information. Might I ask that when you post something like this, you undertake to explain yourself? What sort of sensible response is possible to someone who says nothing beyond “wrong, wrong, wrong”? Why did you post this? ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [December 8, 2005, 5:45am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/89 "2005-12-08T05:45:55Z")
> [@Giraffe](#): > > As I just posted, pretend there is no air resistance. Also, let’s say his wheel bearings are frictionless. Is it possible for the train to move in such a way that the rocketman stays stationary with respect to the ground? [Pretty pictures](http://rapidshare.de/files/8799403/train.pdf.html) Yes, basically what happens is that the wheels are continously accelerated. Its impossible to do continously becuase you will bump up against the limits of the speed of light and the physical charecteristics of the materials you are using. What is necessary for the rocket to not move in relation to a person standing on the ground is that the force due to friction between the train and the wheel must counteract the force from the rocket. Lets call the force between the train and the wheel Ft and the rocket force Fr. Now our first relationship is Ft=Fr becuase we want no motion. To find out what Ft is and consequently Atrain we need to look at the wheel. The angular acceleration of the wheel is equal to Ft_Rwheel/Iwheel. We need to look at the point of contact between the wheel and the train becuase those two accelerations must be equal becuase the wheel does not slip. The relationship between linear acceleration and angular acceleration is alpha=Atrain/Rwheel. Now we have three equations Ft=Fr, alpha=Ft_Rwheel/Iwheel and alpha=Atrain/Rwheel. Combining them we end up with Atrain=Fr\*Rwheel^2/Iwheel. > [@borschevsky](#): > > I agree with this, depending on what is meant by “speed of the wheels”. Say the plane does take off; when it does, it has some positive speed X relative to the ground. The treadmill is going in the other direction at some speed Y, so the wheels are turning at X+Y. But the OP said that the treadmill always goes at the same speed as the wheels, so this is impossible. That’s unless you allow Y to be infinite, so that X+Y = Y. Its nonsensical to say that the treadmill is moving with the same speed as the wheels. What you are basically saying is that the units for rotation are a distance/time when they are in fact degree/time. The only thing that makes sense is to assume that the linear speed of the wheels are what they are talking about. That speed is the same as the planes speed forward. > [@Xema](#): > > Might I ask that when you post something like this, you undertake to explain yourself? > > What sort of sensible response is possible to someone who says nothing beyond “wrong, wrong, wrong”? Why did you post this? To hopefully ward off anyone being confused by an incorrect answer. Look guys, I’ve made a big post and a lot of pretty pictures about the question in the OP. My answer is right and the plane will take off. ---
**Author:** ![Xema](https://avatars.discourse-cdn.com/v4/letter/x/9de053/32.png) [@Xema](https://boards.straightdope.com/u/Xema)\ **Post date:** [December 8, 2005, 5:55am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/90 "2005-12-08T05:55:00Z")
> [@treis](#): > > To hopefully ward off anyone being confused by an incorrect answer. Saying “wrong, wrong, wrong” clears up confusion and corrects errors? I think you should post a specific and detailed explanation of what you believed to be wrong, and why. ---
**Author:** ![borschevsky](https://avatars.discourse-cdn.com/v4/letter/b/97f17d/32.png) [@borschevsky](https://boards.straightdope.com/u/borschevsky)\ **Post date:** [December 8, 2005, 5:58am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/91 "2005-12-08T05:58:43Z")
> [@treis](#): > > Its nonsensical to say that the treadmill is moving with the same speed as the wheels. What you are basically saying is that the units for rotation are a distance/time when they are in fact degree/time. The only thing that makes sense is to assume that the linear speed of the wheels are what they are talking about. That speed is the same as the planes speed forward. Yes, I agree it’s nonsensical, that’s where the infinity comes from. I thought it might be meant this way because the OP specifically said the speed of the wheels as opposed to the speed of the plane. Also because there’s this big discussion, and if the OP means it your way, then yeah, the plane just takes off, no problem. ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [December 8, 2005, 5:59am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/92 "2005-12-08T05:59:04Z")
I posted the correct solution in my previous post. ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [December 8, 2005, 6:02am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/93 "2005-12-08T06:02:50Z")
> [@Xema](#): > > Okay, let’s pick a sample case: No wind. Piper Cub (which flies happily at 45mph), is rolling north at 50mph, along with its wheels. Treadmill is thus moving south at 50mph. > > …The wheels are spinning at a rather brisk clip (equivalent to rolling at 100 mph) but that’s no big deal. The example above meets all your specifications and has the plane moving forward at comfortable flying speed. Your Piper Cub is moving north at 50 mph relative to the ground. Your treadmill is moving south at 50 mph relative to the ground. The wheels of the Piper Cub therefore have a linear speed of 100 mph relative to their axles. The problem with this is that the OP states that “the conveyer belt is designed to exactly match the speed of the wheels at any given time.” Assuming that the speed of the wheels means “speed of the wheels with respect to their axles”, then in your situation above, the speed of the wheels is 100 mph, while the speed of the treadmill is only 50 mph. ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [December 8, 2005, 6:03am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/94 "2005-12-08T06:03:36Z")
> [@borschevsky](#): > > Yes, I agree it’s nonsensical, that’s where the infinity comes from. No its nonsensical becuase its not the correct units. Its like if I asked you how much you weighed and you told me 3 feet. The units in your answer make no sense in relation to the question I asked. Its the same thing in this situation. I ask you how fast the treadmill is moving and you tell me 3 degrees/second. Its a nonsensical answer becuase a treadmill moves with units of distance/time not degree/time. ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [December 8, 2005, 6:05am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/95 "2005-12-08T06:05:01Z")
> [@robby](#): > > The wheels of the Piper Cub therefore have a linear speed of 100 mph relative to their axles. > > The problem with this is that the OP states that “the conveyer belt is designed to exactly match the speed of the wheels at any given time.” > > Assuming that the speed of the wheels means “speed of the wheels with respect to their axles”, then in your situation above, the speed of the wheels is 100 mph, while the speed of the treadmill is only 50 mph. 😕 The wheels of the plane aren’t moving in relation to the axles. ---
**Author:** ![Manduck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manduck/32/256_2.png) [@Manduck](https://boards.straightdope.com/u/Manduck)\ **Post date:** [December 8, 2005, 6:08am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/96 "2005-12-08T06:08:08Z")
> [@Xema](#): > > Okay, let’s pick a sample case: No wind. Piper Cub (which flies happily at 45mph), is rolling north at 50mph, along with its wheels. Treadmill is thus moving south at 50mph. Surely when it says the treadmill is moving the same speed as the wheel, that refers to the speed at which the wheel is rolling, i.e., a point on the wheels circumference is moving at 50 mph around the axle. With the treadmill also moving 50 mph, the plane has a net speed of zero. Otherwise the wheel would have to be skidding along the treadmill, and presumably that isn’t allowed (because it would make the problem trivial). > [@](#): > > Why not? The wheels are spinning at a rather brisk clip (equivalent to rolling at 100 mph) but that’s no big deal. The example above meets all your specifications and has the plane moving forward at comfortable flying speed. but if the wheel is rolling at 100mph, the treadmill has to also be going 100 mph. ---
**Author:** ![The\_Controvert](https://avatars.discourse-cdn.com/v4/letter/t/dec6dc/32.png) [@The\_Controvert](https://boards.straightdope.com/u/The_Controvert)\ **Post date:** [December 8, 2005, 6:08am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/97 "2005-12-08T06:08:43Z")
> [@Xema](#): > > The example above meets all your specifications and has the plane moving forward at comfortable flying speed. No, it doesn’t. The OP says the treadmill matches the speed of the wheels (100), not the speed of the plane (50). ---
**Author:** ![Manduck](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/manduck/32/256_2.png) [@Manduck](https://boards.straightdope.com/u/Manduck)\ **Post date:** [December 8, 2005, 6:10am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/98 "2005-12-08T06:10:53Z")
> [@treis](#): > > No its nonsensical becuase its not the correct units. Its like if I asked you how much you weighed and you told me 3 feet. The units in your answer make no sense in relation to the question I asked. Its the same thing in this situation. I ask you how fast the treadmill is moving and you tell me 3 degrees/second. Its a nonsensical answer becuase a treadmill moves with units of distance/time not degree/time. If the circumference of the wheel is 10 feet, and the wheel rotates once per second, it’s rolling at 10 feet per second. No problem at all. ---
**Author:** ![robby](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robby/32/11048_2.png) [@robby](https://boards.straightdope.com/u/robby)\ **Post date:** [December 8, 2005, 6:20am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/99 "2005-12-08T06:20:04Z")
> [@treis](#): > > 😕 The wheels of the plane aren’t moving in relation to the axles. The wheels have some definite angular velocity with respect to their axis of rotation, right? A given point on the outer edge of the wheel has a tangential velocity equal to the angular velocity times the radius. The magnitude of this tangential velocity is the tangential speed of the wheel at that point, relative to the axis of rotation, which is the axle. This what I meant by the speed of the wheels relative to the axle. ---
**Author:** ![treis](https://avatars.discourse-cdn.com/v4/letter/t/bc79bd/32.png) [@treis](https://boards.straightdope.com/u/treis)\ **Post date:** [December 8, 2005, 6:20am UTC](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567/100 "2005-12-08T06:20:55Z")
> [@Manduck](#): > > If the circumference of the wheel is 10 feet, and the wheel rotates once per second, it’s rolling at 10 feet per second. No problem at all. Right that makes sense becuase you are talking about the translational velocity of the wheel instead of its rotational. But when you say its spinning at X+Y that entails units of rotation. It does make sense to assume they are talking about translational velocity (its what I did). Then there is no problem becuase if your plane is moving 200 mph to the left relative to the ground the treadmill is moving 200 mph to the right relative to the ground and your wheels are spinning at the rate of w=400 mph/r. [Previous page](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567.md?page=4) [Next page](https://boards.straightdope.com/t/will-a-plane-on-a-treadmill-take-off/334567.md?page=6)