6.2 Types of collisions  (Page 4/4)

Two balls of clay, $200g$ each, are thrown towards each other according to the following diagram. When they collide, they stick together and move off together. All motion is taking place in the horizontal plane. Determine the velocity of the clay after the collision.

1. Analyse the problem

   This is an inelastic collision where momentum is conserved.

   The momentum before = the momentum after.

   The momentum after can be calculated by drawing a vector diagram.

2. Calculate the momentum before the collision
   $$\begin{array}{c}\hfill p_1\left(\mathrm{before}\right)={\mathrm{m}}_1{\mathrm{v}}_{\mathrm{i}1}=(0,2)\left(3\right)=0,6\mathrm{kg}·\mathrm{m}\mathrm{cdot}{\mathrm{s}}^{-1}\mathrm{east}\\ \hfill p_2\left(\mathrm{before}\right)={\mathrm{m}}_2{\mathrm{v}}_{\mathrm{i}2}=(0,2)\left(4\right)=0,8\mathrm{kg}·\mathrm{m}\mathrm{cdot}{\mathrm{s}}^{-1}\mathrm{south}\end{array}$$
3. Calculate the momentum after the collision.

   Here we need to draw a diagram:

   

   $$\begin{array}{ccc}\hfill p_{1+2}\left(\mathrm{after}\right)& =& \sqrt{{(0,8)}^2+{(0,6)}^2}\hfill \\ & =& 1\hfill \end{array}$$
4. Calculate the final velocity

   First we have to find the direction of the final momentum:

   $$\begin{array}{ccc}\hfill tan\theta & =& \frac{0,8}{0,6}\hfill \\ \hfill \theta & =& 53,{13}^{\circ }\hfill \end{array}$$

   Now we have to find the magnitude of the final velocity:

   $$\begin{array}{ccc}\hfill p_{1+2}& =& m_{1+2}{v}_f\hfill \\ \hfill 1& =& (0,2+0,2)v_f\hfill \\ \hfill v_f& =& 2,5m·s^{-1}53,{13}^{\circ }\mathrm{South}\mathrm{of}\mathrm{East}\hfill \end{array}$$

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Collisions

1. A truck of mass 4500 kg travelling at 20 m $·$ s ${}^{-1}$ hits a car from behind. The car (mass 1000 kg) was travelling at 15 m $·$ s ${}^{-1}$ . The two vehicles, now connected carry on moving in the same direction.
   1. Calculate the final velocity of the truck-car combination after the collision.
   2. Determine the kinetic energy of the system before and after the collision.
   3. Explain the difference in your answers for b).
   4. Was this an example of an elastic or inelastic collision? Give reasons for your answer.
2. Two cars of mass 900 kg each collide and stick together at an angle of 90 ${}^{\circ }$ . Determine the final velocity of the cars if car 1 was travelling at 15m $·$ s ${}^{-1}$ and car 2 was travelling at 20m $·$ s ${}^{-1}$ .

   

Tiny, violent collisions

Author: Thomas D. Gutierrez

Tom Gutierrez received his Bachelor of Science and Master degrees in Physicsfrom San Jose State University in his home town of San Jose, California. As a Master's student he helped work on a laser spectrometer at NASA AmesResearch Centre. The instrument measured the ratio of different isotopes of carbon in CO ${}_2$ gas and could be used for such diverse applications as medical diagnostics and space exploration. Later, he received his PhD inphysics from the University of California, Davis where he performedcalculations for various reactions in high energy physics collisions. He currently lives in Berkeley, California where he studies proton-protoncollisions seen at the STAR experiment at Brookhaven National Laboratory on Long Island, New York.

High Energy Collisions

Take an orange and expand it to the size of the earth. The atoms of the earth-sized orange would themselves be about the size of regular oranges andwould fill the entire “earth-orange”. Now, take an atom and expand it to the size of a football field. The nucleus of that atom would be about thesize of a tiny seed in the middle of the field. From this analogy, you can see that atomic nuclei are very small objects by human standards. They areroughly ${10}^{-15}$ meters in diameter – one-hundred thousand times smaller than a typical atom. These nuclei cannot be seen or studied via anyconventional means such as the naked eye or microscopes. So how do scientists study the structure of very small objects like atomic nuclei?

The simplest nucleus, that of hydrogen, is called the proton. Faced with the inability to isolate a single proton, open it up, and directly examinewhat is inside, scientists must resort to a brute-force and somewhat indirect means of exploration: high energy collisions. By colliding protonswith other particles (such as other protons or electrons) at very high energies, one hopes to learn about what they are made of and how they work.The American physicist Richard Feynman once compared this process to slamming delicate watches together and figuring out how they work by onlyexamining the broken debris. While this analogy may seem pessimistic, with sufficient mathematical models and experimental precision, considerableinformation can be extracted from the debris of such high energy subatomic collisions. One can learn about both the nature of the forces at work andalso about the sub-structure of such systems.

The experiments are in the category of “high energy physics” (also known as “subatomic” physics). The primary tool of scientific exploration inthese experiments is an extremely violent collision between two very, very small subatomic objects such as nuclei. As a general rule, the higher theenergy of the collisions, the more detail of the original system you are able to resolve. These experiments are operated at laboratories such asCERN, SLAC, BNL, and Fermilab, just to name a few. The giant machines that perform the collisions are roughly the size of towns. For example, the RHICcollider at BNL is a ring about 1 km in diameter and can be seen from space. The newest machine currently being built, the LHC at CERN, is a ring 9 km indiameter!

Casestudy : atoms and its constituents

Questions:

1. What are isotopes? (2)
2. What are atoms made up of? (3)
3. Why do you think protons are used in the experiments and not atoms like carbon? (2)
4. Why do you think it is necessary to find out what atoms are made up of and how they behave during collisions? (2)
5. Two protons (mass $1,67\times {10}^{-27}$  kg) collide and somehow stick together after the collision. If each proton travelled with an initial velocity of $5,00\times {10}^7\mathrm{m}·{\mathrm{s}}^{-1}$ and they collided at an angle of 90 ${}^{\circ }$ , what is the velocity of the combination after the collision. (9)