It was once argued that rockets would never work because of Newton’s 3rd law: If there is nothing to push against in space, then there will be no way for a rocket to make itself go. (Certainly, your car won’t work in space, since there are no roads for the tires to push against.) So how does a rocket work? Answer the following questions:
The first couple of questions here are easy. With no motion in the beginning, all velocities are zero, which means the total momentum has got to be zero. That said, the momentum ten minutes later or ten years later (if this system isn't affected by anything external) will still be zero. We know this because momentum is conserved when no external impulses are involved.
If momentum is conserved, then any momentum that is going one direction will be exactly cancelled by some momentum in the opposite direction. If the mass of the banana is 1/1000 of the mass of the boat, then for the momentum to remain zero, the boat must "recoil" in the opposite direction of the banana at 1/1000 of the banana's speed. (In this way, the m*v of the boat is exactly the same, but in the opposite direction, as the m*v of the banana.) Thus, with each banana thrown, the boat will move at an additional 10/1000 m/s or .01 m/s. This is not very fast . . . yet.
If you throw out more bananas, then each one will boost the velocity of the raft by .01 m/s. If you want to move at 1 m/s, then you need to throw 100 bananas, since 100 * .01 m/s = 1 m/s. That's a lot of bananas, and probably not the best use of your food supply. But, it gives you the idea as to how rocket propulsion works. If you could throw the bananas much much faster, this would be more efficient, of course.
One other thing that this problem doesn't take into account is how much the mass of the boat changes each time you toss one banana overboard. Since each banana is relatively small, this isn't a big deal in the short term, but you can imagine that after you've thrown hundreds of bananas off the boat, the boat would be a lot less massive, and so it would be able to move faster and faster with each banana toss. We didn't factor this into the problem because 1. It made it harder, and 2. It wasn't too important, since you didn't change the mass of your raft too dramatically over the short term.
(Note: Rockets use the exhaust of fast moving gases instead of bananas. These gases do not have very much mass, but they have very high velocity. (Even higher than our bananas!) The rocket and its exhaust have a total momentum of zero, just like the raft and its bananas.)
A demonstration of a rocket throwing out bananas (pellets) can be found on the momentum animations page. You can vary the mass of the rocket, the mass of the thrown pellets, and the velocity of the pellets.
Two rivals (a chemist, Dr. Seager, and a physicist, Dr. Carroll) stand on a frictionless sheet of ice out in the middle of the Great Salt Lake. They are stranded for days. Finally, Dr. Seager gets tired of Dr. Carroll’s lectures and gives the physicist a push. As the physicist slides off the ice sheet, he laughs at the chemist’s foolishness. Explain why the chemist made a mistake, emphasizing Newton’s Laws and/or conservation of momentum.
The easiest way to explain this is via conservation of momentum. The physicist recognizes the fact that the two persons represent one closed system. If both are motionless, then the initial momentum is zero. If one person starts moving one way as a result of something internal to this system (a push, explosion, etc.), then something has to move the other way in order for momentum to still equal zero. Both scientists will end up wet. Perhaps the physicist is laughing because he is happy that he will get to do an experiment in buoyancy, floating in the salty water.
This can also be explained with Newton's 3rd law: one good push deserves another in the opposite direction. When the chemist pushes on the physicist, the chemist also receives an equal "push" in the opposite direction. Without friction to hold him down, he accelerates and slides off the opposite edge of the ice sheet.
Two balls (red and green) of the same mass are dropped from an identical height. The red ball bounces back upwards, but the green ball hits the ground and stays there. Explain which ball had the greatest amount of force exerted on it, and explain how you know. (You probably want to think about impulse and changes in momentum.) Assume that the ground exerts a force on each ball for the same length of time.
In class we demonstrate a bouncy ball bouncing off the table and contrast it with a lump of clay with the same mass that simply sticks to the table. One way of looking at it is to consider each object's change in motion, or acceleration. The bouncy ball has about twice as much change in its motion, since it turns around and starts moving upwards, as opposed to just coming to a stop. Since its acceleration is twice as great, then the force applied must also be about twice as great.
Another way to talk about this is via momentum. The happy/red ball's momentum changes the most, so its impulse must be greater. An impulse is proportional to the force acting during the collision, so this ball's force must be greater (since the times of contact is the same for each impulse).
Elevator catastrophe
You are standing on a scale inside of an elevator. (Many people are staring at you because you look like some kind of eccentric scientist.) At home, your scale tells you that you weigh 100 pounds. In the elevator, explain what the scale would read (and why!) for the following situations. (In some cases you don’t have enough information to say exactly what the scale reads, but you should be able to explain whether it reads more or less than the original 100 pounds.)
Rocket woman
A person builds a large rocket booster and straps it to her back. Assuming that the Earth is a giant sphere that is rotating on an axis that goes through the North and South Poles, explain from where and in what direction our zealous space traveler should launch herself in order to attain the greatest speed in space. (A sketch might make your answer more clear.)
This potential astronaut should launch herself from the equator (where the rotational speed is greatest), parallel to the horizon and in the direction of the Earth's rotation. (Yes, a sketch would be helpful.) By doing this, the launched person already has this velocity working in her favor (due to inertial properties and Newton's first law -- already in motion and thus will continue in this state of motion) and won't have to use extra boost from the rocket to attain such a velocity.
Born to be wild
Adam, sitting atop his trusty bicycle, pulls up to a stop light. When the light turns green, he pedals the bike and notices that he is going faster than the high powered Corvette that was parked next to him! Assuming that Adam does not have super-human strength (i.e.: force), explain this.
By Newton's second law, the amount of acceleration that an object experiences is not only proportional to how much force is acting on it, but to how little mass the object has. That is, the smaller the mass, the more easily large accelerations are attained. In this case, the ratio of my mass to the car's mass must be smaller than the ratio of my force to the car's force. (This is the most concise way I can say this -- real "English" answers would be acceptable, or showing this in an equation form (a=F/m, etc.) is great too.) Note: I do eventually lose the race. Why?
You are standing on a scale inside an elevator. At home, the scale would read 100 lbs. Currently, the scale reads 120 lbs. There could be two different reasons for this. Explain exactly what 2 motions the elevator could be experiencing. Be sure to describe the velocity and the acceleration of the elevator, and why these make you feel heavier than you really are.
If the scale reads more than what you are accustomed to, it means that the scale is pushing upwards on you with more force than what is required to just hold your weight. In this case, the scale is pushing up on you with 20 lbs more force than is necessary to just hold your weight. So, since this means there is extra pushing in the upward direction (in other words, there is a net force in the upward direction), you must be accelerating in the upward direction. This could mean one of two things: (1.) You could be moving upwards, going faster and faster (or going from rest to faster upward motion; or (2.) You could be moving downward, but more and more slowly, coming to rest. Either of these situations is a change in motion in the upward direction, or an acceleration in the upward direction. Either requires the thread of the elevator to pull upwards, and in turn requires the floor and scale to push upward on you.
Let’s imagine that the contents of the entire universe go away, leaving only you and your dog. The two of you are floating out in space. You are three times as massive as your dog. Each of you feels a gravitational force towards one another, just as Newton’s law of gravitation describes.
You and your dog feel exactly the same force towards one another. This is Newton's 3rd law. You must remember that, even though the two masses are different, gravity is always an interaction of two masses. One mass does not gravitational force make (or something like that . . . ). So, the two pull on each other, and they do so equally, because the masses involved work together. (However, even though the forces are the same, the resulting accelerations are different. This is why your dog will change is motion more rapidly (by 3 times) than you will yours. For the same reason, the Moon goes around the Earth more dramatically than the Earth goes 'round the Moon, even though the two objects feel exactly the same force towards one another.)
If your astro-dog doubles his mass, the gravitational force also doubles. (See Newton's law of gravitation, which shows the force being proportional to the masses involved.) So, you would feel 16 Newton's of force, as would your dog.
If your dog were twice as far from you (at his original mass), the force would decrease by a factor of 4. This is because the distance not only decreases the force, but the distance is related to the force as an inverse square. In other words (looking at the law of gravitation again), the factor that the force of gravity is reduced by is the distance squared, or that factor of 2, squared, which is 4. (This is one of those things that is much nicer to see in an equation form than in a paragraph form.)
You are standing in a giant, glass school bus that is moving down a perfectly level and smooth road at a constant velocity of 20 m/s. You have a stuffed dinosaur with you (for comfort and companionship). If you toss your stuffed purple dinosaur straight into the air, describe his motion from:
You can use a sketch to describe the motion, but also explain (in words) what is taking place.
When you throw Barney up into the air, you simply see him go straight up and come straight back down. Since the two of you are in the same boat/bus/container (also known as a "reference frame"), the two of you have the same motion to begin with. You don't have to even know what this motion is (usually we don't!), but you can still count on Barney falling right back into your arms and giving you a great big hug, so long as the container you're in doesn't accelerate (change its motion) sometime during the flight of the dinosaur.
From an outsider's perspective, Barney not only goes up and down, but he is also going from left to right at 20 m/s. The observer sees a projectile path of Barney. Horizontally, Barney moves right along with the bus and with you, and vertically he falls at exactly the same rate of acceleration (gravity's) that you witness him fall at. Thus, if the two of you timed how long it took Barney to go up and down, you would both come up with the same answer.
(Towards the end of the course, we'll cover special relativity, which notes that for very fast speeds, the timing conducted by the two different observers would actually be noticeably different. Very odd, you might think, though some could also argue that it is very odd that the two observers would ever measure the same time for the two observations. We can worry about all this madness later.)
While in midair, Barney feels only one force (if we ignore air resistance/friction): gravity. He feels this same net force for the entirety of his flight, so he should feel the same way during the entire flight, now matter if he is going up, stopped, or coming back down! Since nothing is pushing back up on him, he has the sensation of apparent weightlessness. This happens in our scenario of elevators with broken threads, and it also was experienced by a cup of water with holes in it that was thrown up into the air in class. (When the cup was tossed into the air, remember what happened to the water that 'naturally' gets pulled out of the holes of the cup when the cup sits still?)
How on Earth?
It is described how the Earth and other planets go about the Sun in elliptical paths; and it is also described how the cause of this can be gravity. This is all fine. But, why don't we fall off the Earth if it is moving and spinning at such an outrageous pace? (Hint: The answer is not gravity.)
It is such a wonderful thing, us not falling off the Earth. But why shouldn't we? Well, why don't you fall off an airplane, car, train, etc. while these are moving? Because you are already moving with them Once an object is in motion, it is natural for such motion to continue (Newton's first law). So, since we aren't really sitting still, but are instead moving at the same speed and rotation as the Earth, we continue with this motion and thus continue standing on the same place on the same planet.
It is also true that we are going in circles, and our natural motion is in straight lines. This part of the motion does require gravity. However, even the motion that we are doing in circles is never very curved, so gravity does well more than is required to keep us from flinging ourselves off the planet. It is true that a very very precise scale that you stand on at the equator would show an-ever-so-slightly lower value than the same scale at the North Pole, but it is not even close to anything that you notice.