It’s a Mechanics assignment. First year university course. We are looking at a collision between two identical atoms that form a molecule in the process. We use both analytical methods, such as conservation laws, and numerical methods to study the motion of the molecule. We assume that each atom has mass m and can be considered as a point particle. We assume that all external forces are so small in relation to the forces between the atoms that we do not have to take them into account. We first look at a one-dimensional central collision between the atoms. Atom A moves along the x-axis at velocity vo=vo*i. Atom B is at rest in origin. Atoms do not interact with each other, that is, no forces act between them until they come into contact. After the collision, the atoms are attached to each other. See Figure 1 for an illustration of the collision. (a) Find the velocity of the center of mass of the system consisting of the two atoms both before and after the collision. Explain the procedure and justify your answer. (b) Compare the energy that the system has before and after the collision and discuss the result. Now we try to make the model for the collision more realistic. Instead of the atoms attaching to each other instantly by contact, we model the force between the atoms as a spring force that follows Hooke’s law. When atom A reaches position xA = -b, the atoms are connected together through a spring with a spring constant k and equilibrium length b. We assume that the mass of the spring is very small and therefore negligible in relation to the mass of the atoms. The atoms remain connected to the spring in the further motion. See Figure 2 for an illustration. (I will add figures as attachments) (c) What is the velocity of the center of mass immediately after the atoms are connected to the spring? Compare again the energy that the system has before and after the connection and discuss. (d) We describe the position of the atoms using the coordinates xA and xB. Write down an expression for the forces F?AB and F?BA from atom A on atom B and vice versa as a function of the positions. Check that the forces have the correct sign. Find the expressions for the accelerations of the two atoms A and B. What are the initial conditions? (e) Write a program that plots the positions of the atoms as a function of time for the first 2 seconds after they are connected. Use the parameters m = 0.1, k = 20, b = 0.2, v0 = 1.0 and ?t = 0.001 (in appropriate units). Plot the positions of atoms A and B as a function of time in the same figure. (f) Calculate the maximum and minimum distance between the atoms analytically and check that the result is consistent with the numerical calculation. So far, we have only looked at a one-dimensional collision. Now we’ll look at a non-central collision. We start again with a simplified model where the coupling is instantaneous (without springs). Atom A moves in the x-direction at a distance y = b from the x-axis at speed v0 = v0*i. Atom B is at rest in origin. No forces act between the atoms until atom A reaches the positio r? = b*j. At this moment, the atoms are connected, and the atoms remain at a fixed distance b from each other. We assume that the connection between the atoms in the molecule is rigid (ie without feathers). The system of the two interconnected atoms now has an angular velocity ?? = -?*k. See Figure 3 for an illustration. (attachment) (g) Find the velocity to the center of mass to the molecule and the angular velocity around the center of mass after coupling. Explain the procedure and justify your answer. We try to make the model more realistic again by connecting the atoms together through a massless spring with a spring constant k and equilibrium length b. Atom A moves again in the x-direction at a distance y = b from the axis at velocity v0 = v0*i, while atom B is i ro i origo. The coupling takes place when atom A reaches the position ?r = b*j. The atoms remain connected through the spring afterwards. (h) Show that the expressions for F?AB and F?BA (the force from A on B and B on A), can be written as … (equation 1 in attachments). using the distance vector ??r = ?rB – ?rA, the length of the distance vector | ??r |, the equilibrium length b and the spring constant k (hint: since ?rB = ?0 is ??r = -?rA). Check that the force vectors F?AB and F?BA have the correct direction. (i) Write about the program in Matlab or python to calculate the motion of the two atoms after the interconnection in the non-central collision. Plot the position (x (t), y (t)) of both atoms in the same figure, and describe the motion. You can use the same parameters as in task (s). What does the movement to the mass center look like? I will add the document with the description as attachment. It also has the figures and equations. It does not have to be 3 pages, but since I had to choose a number.
