Show that the kinetic energy of two particles with mass m1 and m2 split into the energy of the centre of mass and the kinetic energy of relative motion.
Huh? This makes no sense.
To show how the kinetic energy of two particles with masses m1 and m2 splits into the energy of the center of mass and the kinetic energy of relative motion, we can start by defining the center of mass of the system.
The center of mass (COM) is a point in space that represents the average position of all the particles in the system. It is given by the equation:
COM = (m1 * r1 + m2 * r2) / (m1 + m2)
where m1 and m2 are the masses of the particles, and r1 and r2 are their respective positions.
Now, let's consider the velocities of the two particles. The velocity of a particle is given by:
v1 = d(r1)/dt
v2 = d(r2)/dt
By taking the derivative of the equation for the center of mass, we can find the velocity of the center of mass (Vcom):
Vcom = d(COM)/dt = (m1 * v1 + m2 * v2) / (m1 + m2)
Now, let's calculate the kinetic energy of the system. The kinetic energy of an object is given by:
K = (1/2) * m * v^2
where m is the mass and v is the velocity.
The total kinetic energy (K_total) of the system is the sum of the kinetic energies of the two particles:
K_total = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
We can rewrite this in terms of the velocities of the center of mass and relative motion. The velocity of relative motion (Vrel) is defined as:
Vrel = v1 - Vcom = v1 - (m1 * v1 + m2 * v2) / (m1 + m2)
Substituting this expression for Vrel into the equation for K_total, we get:
K_total = (1/2) * m1 * (v1 - Vcom)^2 + (1/2) * m2 * v2^2
Expanding and simplifying this equation further, we obtain:
K_total = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 - (m1 * v1^2 + m2 * v2^2) / (2 * (m1 + m2))
Now, let's evaluate the expression in the parentheses:
(m1 * v1^2 + m2 * v2^2) / (2 * (m1 + m2))
This term represents the kinetic energy of the center of mass (Kcom), obtained by substituting the velocities and masses:
Kcom = (m1 * v1^2 + m2 * v2^2) / (2 * (m1 + m2))
Finally, substituting Kcom back into the equation for K_total, we find:
K_total = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 - Kcom
Therefore, the total kinetic energy of the system can be expressed as the sum of the kinetic energy of the center of mass and the kinetic energy of relative motion, where Kcom represents the energy of the center of mass and (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2 represents the energy of relative motion.