Collisions of hard spheres

these are collisions in two dimensions in this experiment the student can observe for himself that momentum is a vector quantity and he can determine that it is the vector sum of the momentum not the arithmetical sum that is conserved this idea will only become really clear to him by analyzing two-dimensional collisions like these this is the apparatus with which we made those collisions in two Dimensions this is a standard lab kit if you look from this side over here you will see that just before and after the moment of impact the movement is in a horizontal plane Beyond this point gravity makes the experiment seem three-dimensional but a careful choice of observations reveals the Collision to be as two-dimensional as you saw it in the plan view the incident sphere gets its velocity when it rolls down this ramp and collides with the target sphere out here away from the edge of the ramp the target sphere is supported on top of this set screw which in turn is held in place by this pivot arm directly below the location of the target sphere when the Collision took place hangs the plumbob close to the paper the tracing paper is placed on top of carbon paper so that when the sphere is hit they leave marks here is the point above which the last Collision took place the target sphere hit over here at exactly the same same time the incident sphere hit here thus we can determine the horizontal distance traveled and the direction of each sphere before they hit the floor this experiment ideally supplements the student study of momentum and the conservation of momentum in the text in previous experiments he has also learned learned that the momentum of an object is equal to its mass time its velocity he has learned about projectile motion the fact that objects projected from the same height at different horizontal velocities reach the floor at the same time now he has also learned that the horizontal component of the Velocity remains unchanged now let's see here and here down here it's the same and the same so that the horizontal distance traveled by the object before it hits the floor is proportional to its horizontal velocity this fact is the foundation on which the whole experiment is built earlier in the course the student has learned something about vectors but this is his first real chance to do an experiment that involves the use of vectors the experiment will show him that direction is important that momentum is not a scaler it is a vector quantity and through the experiment the student will see that momentum is conserved as a vector sum not as a scalar or arithmetical sum this is the prime objective of this experiment the apparatus is not too difficult to assemble bend the base plate about like this and clamp it to the edge of the table if the table has a lip on it you could place a block of wood under the base the end of the base should be about 7 to 8 cm above the table there Center the ruler on the plate and snap the ruler into place to begin with we would like to measure the velocity of the incident sphere if we are to accurately measure this velocity before the sphere collides by the method used in this experiment we must make sure that the sphere is well clear of the ramp before it collides for when a sphere rolls over an edge like this the edge exerts a force on the sphere a horizontal force it exerts this Force as long as the sphere is in contact with the edge therefore at a point just one radius out from the edge the sphere will be well clear of of this force and indeed on its own to mark this point we'll set up our Plum line fold the end of the string over the small stick and wedge it into the hole in the bottom of the set screw fasten the arm and pivot it so that a sphere set on The dimple on top of this set screw will just clear the edge of the ramp one radius out the plum Bob should be adjusted so that it hangs 1 or 2 cm above the paper like that now let's remove the weights and this old piece of paper now you can see the four sheets of carbon paper taped together and place carbon side up here are four sheets of onion skin paper taped together and we'll place them on top of the carbon paper and Center it so that the plum Bob Falls along the midpoint of this side weigh down the paper so that it doesn't move during the experiment now we'll mark this point and I guess we're ready to take some d dat swing the pivot arm to the side and we'll measure the incident velocity for this experiment I'll always release the sphere from about the 25 cm Mark we must release it lightly so that the spring of the ramp does not introduce an error let's try it there see the mark and again [Applause] now I'll draw a circle around this group of about 10 points and you can see that all the points lie in an area about 2 cm in diameter the line drawn from the plum Bob point to the center of this distribution of points represents the velocity of the incident sphere and in this case it is 25.9 CM in our unit of time that velocity times the mass of our sphere is equal to its momentum the sphere in this experiment is a 1/2 in steel ball bearing its mass is 8.3 G thus the momentum is equal to 8.3 * 25.9 cm per time interval but the time interval throughout the experiment is constant so let me call this one unit of time also let me call 8.3 G one unit of mass so that the numerical value of the momentum is just 25.9 now for our collisions we want the motion to take place in a horizontal plane the plane that the incident sphere is in when it is just clear of the edge of the ramp we can fix the height of the target Ball by adjusting this set screw up up or down we turn it so that the incident ball just clears it as it goes over the edge of the ramp there that's about it let's try it it hit so we lower it just a bit and try it again there it cleared now I'll tighten this nut so that the set screw will not move during the rest of the experiment that adjustment has our Target ball at the right height and ensures that the collisions will occur in a horizontal plane now we will be concerned with the position of the Spheres in that plane for a head-on collision we place the target sphere here three radi out from the edge of the ramp so that the incident sphere is again just clear of the ramp when it collides or for a glancing Collision over here a little over one radius from The Edge we can study the whole range from a glancing Collision to a headon one with this equipment but it would require fine adjustment of the target ball for each new position so to simplify the experiment for the student we have determined a satisfactory position away from the edge of the ramp an average position this position is 2 and 1/2 radi of the sphere out from the edge of the ramp 2 and a half radi since this is a mid position extreme right and left positions along the axis for a head-on or a glancing Collision will introduce some error but these are not the really interesting ones now for our first Collision there so that we can remember where these spheres hit we'll mark these points with a pair of numbers and while we're at it the th line is hanging directly below the original position of the target sphere and we'll put a mark at this point the distance from this point to where the target ball hit represents its velocity Vector Now we move the target sphere to a new position along this axis and try it again there now we have obtained the data for four different collisions in two Dimensions now let's see what has happened let's analyze our data in our first Collision a line drawn from the original position of the target sphere to where the sphere hit is a measure of its displacement in our unit time thus it is its velocity Vector at the time of collision the incident sphere was along this line and two radi of the sphere behind the original position of the target ball there we draw a line from that point to where the incident sphere hit the paper and that is its velocity Vector now I want to add this velocity Vector to this one I'm going to construct the parallelogram this is the length of the target spheres velocity vector and the vector sum should lie along that Arc this is the length of the infinite spheres velocity Vector but now we must remember that the Spheres were two radi apart when they collided and so we come back along this Vector to there the vector sum should lie at the intersection of those two arcs there is the target sphere's Vector velocity added to that of the incident sphere and it falls within the distribution of points which represents the initial velocity of the incident sphere to within accuracy of this experiment velocity is concerned now let's do the same thing for the other three collisions and you see that in all four cases when the two velocity vectors are added together the vector sum lies within this distribution of points so what have we found out we have found out that in a collision of two dimensions of two equal Mass spheres the velocity is conserved each had a mass of one unit so that momentum 2 that is mass time velocity must be conserved in these collisions now let's perform the experiment with a Target ball of different Mass this is a glass marble it also has a diameter of 1/2 in and a mass of 2.7 G which is a good deal lighter than the mass of our steel ball bearing and about 1/3 of our unit Mass I'll use the lighter sphere as the target sphere if I used it as the incident sphere it would bounce back and hit the edge of the ramp now I put a new spread of paper on the floor and plotted the incident velocity so I guess we're ready to go there the lighter Target sphere seems to go much farther these vectors represent the velocity of the glass Target sphere these the incident sphere if we add the velocity of the target sphere to that of the incident sphere the vector sum Falls way out here not once but every time so that in these collisions of unequal Mass spheres velocity is not conserved but momentum is our Prime concern here we cannot measure it directly as we have the Velocity momentum is the velocity multiplied by mass so in these collisions of unequal Mass spheres let us now consider the momentum the mass of the incident sphere is one unit so that the length of its momentum Vector is the same as its velocity vector the length of the momentum Vector of the target sphere is only 1/3 that of its velocity since its mass is about 1/3 of our unit Mass there these are the momentum vectors for our Target sphere now let's find the vector sum again we see that momentum is conserved convincing evidence from collisions in two dimensions on simple equipment for