Physics lecture 3 Physics lecture 4
Although Newton had the option of writing down his Second Law of Motion as the familiar , he specifically chose to do it as . The reasons for this, as was usually the case with Newton, were due partly to the thinking of the ``giants'' who came before him and guided his thinking, partly by the scientific beliefs of the day, and mostly as a result of his own meticulously conducted experiments. In this case, the thinking of many of the great scientists of the mid 1600's was guided by the work of Rene' Descartes. Descartes was going the next step beyond thinking about the motion of a single object to consideration of the ``acted upon'' interaction between two objects. The simplest kind of interaction is a collision or impact. Although Descartes' theories on what transpired during collisions were incorrect, one powerful notion from him that proved to be convincing and true was that there should be some conserved quantity during the collision. In other words, some property carried by the objects is unaffected by the collision and is therefore preserved.
The concept of conservation is very powerful since, as we shall see, it provides you with constraints on the behavior of a system. Constraints are readily interpreted in mathematics as resolving unknown quantities, hence you reduce a difficult or unsolvable problem to one that is relatively trivial by reducing the number of unknowns. Most of the progress made in physics from the 1500's to now has been on the basis of principles of conservation. We are about to encounter the first of these principles now.
In 1668, the Royal Society of London requested any scientist who had worked on the subject of collisions to communicate their results. After much discussion of experimental results, two observations stood out. The first was that, in many collisions, a quantity formed by the mass of an object and its velocity was conserved during a collision. This means that if an object with a given initial velocity and mass, say v1i and m1i, is collided with another object of a given (not necessarily the same) velocity and mass, say v2i and m2i then the following mathematical relationship holds:
where the f subscript indicates the velocity of the objects after the collision has occured and i indicates the velocity before the collision occurs. The dark print of the v's (or an arrow appearing over a v) means that the direction as well as the magnitude of the velocity must be considered. In collisions of hard objects people such as the Dutch scientist Christian Huygens (famous for theories on optics) and Christopher Wren (the architect of St. Paul's Cathedral) independently concluded that the product of mass and velocity squared was conserved. In mathematical terms, they found that
The situation was confusing because it appeared as though the relationships found definitely held in some collisions but not in all. Newton cleared up things considerably by doing very careful experiments on collisions between objects made from all kinds of materials. He correctly concluded that the relation
was always obeyed in every kind of collision no matter what the colliding objects were made of or what happened during the collision. However, the relation
holds only in collisions between hard objects. Since the first relation is clearly the more general, Newton developed his second law around the concept of linear momentum defined as the quantity and usually denoted in physics by the letter . Again, the arrow indicates that linear momentum (which we shall just refer to as the momentum from now on) needs to be specified by a direction AND a magnitude or amount. The other parameter, mv³ , is usually denoted in physics by the letter, K, and is called the kinetic energy. We shall return to the concept of energy later, but for now you can examine the consequences of these conservation laws by working with the picture shown below. Here, we allow two objects to collide head-on. You can change the mass or velocity of either object, then watch what happens to the linear momentum and the kinetic energy as you specify to what degree the kinetic energy is to be conserved in the collision. We specify the amount of kinetic energy conservation as the elasticity. 100 percent elasticity means kinetic energy is completely conserved while 0 elasticity means the collision loses the maximum amount of kinetic energy it is possible to lose. You can click on the area where the objects move at any point to stop their motion. Simply click start again when you want the objects to start moving again. Although Newton did not explicitly include the concept of time in his Second Law, Leonhard Euler, a famous mathematician, later wrote down the correct expression for force as
Although Newton did not explicitly include the concept of time in his Second Law, Leonhard Euler, a famous mathematician, later wrote down the correct expression for force as
Thus, we have a mathematical description of the term ``acted upon'' which is fully consistent with any experiment we may conduct (as long as the velocities are considerably less than the speed of light). Notice the subscript ext which indicates that the force acting upon the object whose momentum is changing must be external to the object. No amount of motion within the object, no matter how violent, can change the product of its mass and velocity. Only an external object ``acting upon'' it can do so. In the case where no external force is acting, then we say that the momentum of the object is conserved, i.e. does not change with time. In a collision between two objects 1 and 2, object 1 ``acts upon'' object 2 to change its momentum and object 2, by Newton's Third Law, ``acts upon'' object 1 to change its momentum in a way which is equal in magnitude and opposite in direction to the action of 1 on 2. If we look at object 1 and object 2 as being part of the same system, then the collision between them becomes an internal interaction, there are no external forces, and so the momentum of the system, defined now as the sum of the momentum for 1 and that for 2, is conserved (i.e. the ``action upon'' the system is the sum of the action of 1 on 2 and 2 on 1 and these must cancel by Newton's Third Law). The figure below summarizes these ideas.
Recall that, in cases in which hard, undeformable objects collide, Newton found that the relation
was true in addition to the equation for conservation of momentum. Note that this equation does not involve the directions of the velocities but simply their magnitudes. The quantity is called the kinetic energy or energy associated with motion. The kinetic energy is usually described by the letter K and is expressed in units of Joules. 1 Joule = (1 kg)(1 m² /s² ).
The equation above is a consequence of the conservation of mechanical energy. We classify collisions as either elastic or inelastic depending on whether the collision conserves kinetic energy or not. Collisions between hard objects which rebound off each other are completely elastic. Soft objects which stick together after the collision are termed completely inelastic. Although it was not known in Newton's time, conservation of energy is no less fundamental than conservation of momentum. However, since energy can appear in many different forms, some of which are difficult to observe, it took many years and experiments in many branches of physics and chemistry to prove that energy (as distinguished from kinetic energy, the energy associated with motion) is also conserved in all collisions. In inelastic collisions some of the kinetic energy is absorbed into deforming the colliding objects, some into sticking them together, and the remainder appears as heat.
Now that we have presented two conserving equations, we can go through some classic examples that should prove to you how powerful the constraint of having a conserved quantity can be in solving problems. Remember that momentum is conserved for a system only if no external forces act on the system and that kinetic energy is conserved only if the collision is completely elastic.
SOLUTION: From the figure above, we see that the directions can be defined according to our coordinate axes for each velocity. We have used the convention of labelling initial velocities with the letter u and the common final velocity (since a completely inelastic collision means that the cars stick together after colliding) as . Referring to the Cadillac as vehicle 1 and the Honda as vehicle 2, we note that since there is no friction on the road surface and gravity acts only in the vertical direction, then there are no external horizontal forces if we consider our system to include both cars. In this case we have momentum conserved and the equation describing the collision is
Note that we are careful to include the minus sign for car 2 since it is initially moving in the -x direction. We do not know the direction of the final velocity a priori, but it really doesn't matter since momentum conservation will tell us whether our arbitrary choice of pointing to the right is correct or not. If, after all our calculations, we get a negative value for , then we would conclude that points opposite to the direction we indicate in the figure above, i.e. the cars leave the collision going to the left instead of the right as we show it above. If is positive, then we have correctly chosen the direction for the after-collision case. All that is demanded of us is that we choose some horizontal direction and be consistent with it throughout our calculation. We quickly solve our equation for the unknown to get
As mentioned earlier, the minus sign means that the cars actually go in the -x direction or to the left after the collision. To get the fractional change in kinetic energy, we make use of the definition of kinetic energy:
The fractional change is
So we see that almost all of the energy associated with the motion of the cars turns into some other form of energy. In this case, most of the energy winds up in deforming the cars since automobiles are designed to absorb energy in their fenders and front body parts during a collision.
To get a better realization of what these equations imply, let's consider collisions in which kinetic energy is conserved. If we narrow things still further to collisions in which one of the objects is initially at rest, then we have only three cases to consider. One in which the object in motion is heavier than the stationary object it strikes, one in which the object in motion is lighter than the stationary object, and the finally the case where the moving and stationary objects have equal masses. Maple will easily solve the equations which give us the final velocities in terms of the masses and initial velocity of the originally moving object, but we can intuit what will happen if we just use our experience to consider cases in which the masses of the two object are very different or very similar. The videos below show various balls from different sports in collision in outer space. We use the outer space setting so that we can be free of the effects of gravity, air resistance, surface friction, etc. You should think about what should happen in such a pristine environment, then play the video to see if your intuition matches ours.
First, we consider one billiard ball hitting another: what should you see?
Next, we consider a bowling ball on a baseball: what do you expect happens?
Finally, consider the collisions of two equal mass pool balls: now what happens?
Maple makes short work of some of the classic problems in momentum conservation. Do exercise 3-10 before reading further.
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