Obtain the relation between the total kinetic energy in the CM and Lab frames. Discuss the limiting cases of very large and very small mass for the target.

First, let's define the two frames of reference involved in this problem. - Center of Mass Frame (CM): In this frame, the total momentum is always zero. The coordinate system is defined with its origin at the center of mass of the system. - Lab Frame: This frame is an observer's frame of reference, typically stationary relative to the observer (e.g., the Earth). Now let's identify the system we will be using in this problem, which will be a collision between two particles, one with mass m1 and the other with mass m2.

Let's assume that the velocities of the two particles in the lab frame are V1 and V2. Then the momenta are P1 = m1*V1 and P2 = m2*V2. The total momentum in the lab frame is P = P1 + P2. Now, we change to the center of mass frame. We subtract the velocity of the CM frame v_cm from V1 and V2 to obtain velocities of particles in the CM frame: U1 = V1 - v_cm, U2 = V2 - v_cm. The momenta in the CM frame will be M1 = m1*U1 and M2 = m2*U2. By definition, the total momentum in CM frame must be 0, so we find: M1 + M2 = m1*U1 + m2*U2 = 0

Now we calculate the total kinetic energy (KE) in the lab frame and CM frame. For the lab frame: KE_lab = 0.5*m1*(V1^2) + 0.5*m2*(V2^2) For the CM frame: KE_cm = 0.5*m1*(U1^2) + 0.5*m2*(U2^2)

First, express U1 and U2 in terms of V1, V2, and v_cm: U1 = V1 - v_cm U2 = V2 - v_cm Now substitute these expressions into the KE_cm formula: KE_cm = 0.5*m1*((V1 - v_cm)^2) + 0.5*m2*((V2 - v_cm)^2) Subtract KE_cm from KE_lab to eliminate v_cm: KE_lab - KE_cm = 0.5*m1*(V1^2) + 0.5*m2*(V2^2) - [0.5*m1*((V1 - v_cm)^2) + 0.5*m2*((V2 - v_cm)^2)] Simplify the expression to find the relationship between KE_lab and KE_cm: KE_lab - KE_cm = 0.5*(m1*m2)/(m1+m2) * (V1 - V2)^2

For the limiting cases of very large and very small mass for the target (m2), we discuss how the total kinetic energy is affected in each case. 1. Very large target mass (m2 >> m1): The fraction (m1*m2)/(m1+m2) will approach m1, thus making the quantity (KE_lab - KE_cm) larger. In this case, a large amount of kinetic energy is associated with the internal motion of the system. 2. Very small target mass (m1 >> m2): The fraction (m1*m2)/(m1+m2) will approach m2, thus making the quantity (KE_lab - KE_cm) smaller. In this case, very little kinetic energy is associated with the internal motion of the system. In summary, the relationship between the total kinetic energy in the center of mass frame and the lab frame is given by the expression: KE_lab - KE_cm = 0.5*(m1*m2)/(m1+m2) * (V1 - V2)^2 For very large and very small target mass, we discussed the implications of the kinetic energy being more or less associated with the internal motion of the system.