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Name: Peter
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Question:
There is something that I think is not explained well in the classic Newton Car experiment. In that experiment, a block is shot off the back of a car (using rubber bands) in one direction, causing the car to roll in the opposite direction. This experiment explains Newton's Second Law: orce = Mass x Acceleration So for this experiment: Force (of rubber band) = Mass (of Block) x Acceleration (of Block) It also explains Newton's Third Law of equal and opposite reaction: The action force of the block equals the reaction force exerted on the car. Mass (of Car) x Acceleration (of Car) = Mass (of Block) x Acceleration (of Block) There is one series of trials where the number of rubber bands are increased to create a greater force and the distance the car rolls is measured. With a greater force, the car rolls further. That is understandable. The question come about when the mass of the block is increased. In the experiment, weights are added to the block and the distance the car rolls is measured. As weights are added, the car rolls further. The explanation given is "Newton's second law states that a larger mass causes a larger force." But why is that? My thinking is the force is really caused by the rubber band. That force remains the same. If the mass of the block is increased, the acceleration of the block should decrease according to Newton's Second Law. Acceleration (of Block) = Force (of rubber band) / Mass (of Block) The force, then, on the car, should remain the same. But the experiment shows the car going further with a greater mass of the block. Why does this equation not hold? Mass (of Car) x Acceleration (of Car) = Mass (of Block) x Acceleration (of Block) The results prove that a greater force is being exerted with a larger mass. But, since the force (the rubber band) is the same, it seemingly violates Newton's Second Law. So something else must be going on that is not apparent.



Replies:
Hi Peter,

It is momentum that is conserved, not force, so equating F = -F = m1a1 = -m2a2 is not quite right. You should have m1v1=-m2v2. (where m1 equals mass 1, with 1 meaning the block, m2 is mass 2, say of the car, and v is the velocity). Newton's third law does not mean "conservation of forces", so I think part of the problem is with how you are applying the third law.

Since it is momentum that is conserved, not force, whatever momentum is transferred to the block will acquired by the car (neglecting effects of the bands, etc.). If the mass of the block is increased, and its velocity is held constant, then when the bands are released, the momentum of the block will change (i.e. it moves) increase more. Equally larger momentum will be transferred to the car in the opposite direction (because momentum is conserved), and since the car mass does not change, its velocity will increase, causing it to move farther.

You also run into problems with simplifying assumptions being made about the rubber bands -- are you presuming they have a constant amount of energy stored? A constant velocity? It sounds like you are presuming a constant force -- although I am not sure "real" rubber bands would supply a constant force. So as weight is increased to the blocks, you need to be clear how your rubber bands (real or assumed) are acting. You also do not specify the time/distance for which the force is applied. In short, the problem as you describe it is under-defined in that regard.

Hope this helps,

Burr Zimmerman


Peter,

Mass does not cause acceleration. It resists it. It resists its own acceleration. If you push on a very heavy object, the object resists being accelerated. Because of this, the object pushes back harder. You cannot just choose to push hard on something. It must be able to push back. It is easy to push with ten Newtons of force on a bowling ball. It is almost impossible to do the same to a feather. The greater mass being pushed on the cart resists being accelerated by the rubber bands, pushes back harder on the rubber bands. This in turn allows the rubber bands to push a little harder than before.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College


The missing element in your analysis is the duration of the force, i.e. how long the rubber band is pulling the car forward. You are quite right that, during the first instant of the experiment, when the rubber band is fully extended, the force on the car is the same for heavy and light blocks. Thus, the car's initial acceleration is the same in both cases. But, as you point out, the car goes farther in the heavy-block case than it does in the light-block case.

To understand how this makes sense, you need to remember that what is relevant in determining the car's final speed is not only the instantaneous magnitude of the acceleration, but also its duration (mathematically, it is the integral of the force over time that is proportional to the final velocity). Think about driving an ordinary car: if you push the accelerator down for half a second, the car does not speed up much, but if you hold it down for five seconds, it speeds up quite a bit.

Now, in the case of the light block, the rubber band quickly shoots it away and relaxes, so the duration of the force is short. But, in the case of the heavier block, the band can not fling it away as fast, so the duration is longer.

Douglas Sanford


I had to think a while about what the experimental set-up is to be able to comment on it. First, I will summarize what I understand is happening in the situation, then I will be able to comment on the physics. If my understanding of the set-up is incorrect, obviously you will need to disregard my analysis.

I gather that the situation is this: rubber bands are somehow used to fire blocks of different masses off a cart that is free to roll. As the blocks move in one direction, the cart rolls the other direction. This exemplifies Newton's third law, that the cart and the block, acting on each other by way of the rubber band(s), apply equal-magnitude forces to each other in opposite directions.

When a more massive block is fired off the cart, the cart rolls farther. Why is this? Well, it is not a simple matter of changing a single variable in either of the two Newton's laws. Rather it is a matter of time, momentum, distance, and energy. The difference is basically this: since the block is now more massive, is accelerates less under the force from the rubber band. This means that it takes a longer time to part company with the cart, and that the rubber band is thus pushing on it for a longer time. Newton's third law requires that as long as the block is being pushed by the rubber band, so is the cart, in the opposite direction. So the cart also is pushed by the same (range of) force, and also over a longer time. Since the cart receives its force for a longer time, it spends more time accelerating, and thus reaches a higher final speed.

The heavy block is pushed by the same (range of) force as the lighter block, but over a longer time and a slightly shorter distance. (Why a shorter distance since the rubber band has to move through the same length change? Because the cart is now moving in the opposite direction faster than it was with the lighter block, so the heavy block has not moved as far by the time it leaves the cart.) In a nutshell:

* Both the block and cart receive greater-magnitude impulses (force x time) when the block is massive than when the block is light.
* Both the block and cart experience greater momentum changes when the block is heavy. Since the cart's mass is the same in both instances, that means its speed must be greater.
* When the block is light, it travels farther while the force is applied. This means greater work is done on the light block than on the heavy block, and consequently the light block has more kinetic energy than the massive block.
* When the block is heavy, the CART travels farther while the force is applied. This means that more work is done on the cart when the block is massive than when the block is light. Thus, the cart also gains more kinetic energy when the block is massive than when the block is light.
* The total momentum change of the (cart + block) is zero in both cases. The cart and block undergo equal and opposite momentum changes. The cart's momentum changes the most when the block is massive.
* The total kinetic energy gain is also the same in both cases, because the same amount of potential energy stored in the stretched rubber band becomes kinetic energy of the block and cart. When the block is light, more of the kinetic energy gain is by the block, and less is by the cart, than when the block is massive.

Richard Barrans Jr., Ph.D., M.Ed.
Department of Physics and Astronomy
University of Wyoming



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