Center-of-Mass Frame 2D Harmonic Oscillation: Trajectory of a Puck Under Special Attractive Force

Problem Statement

On an infinitely wide smooth horizontal surface ($xy$xy-plane), there are two pucks A and B treated as point masses.

Between the two pucks, a special attraction mechanism of negligible mass is attached. When the distance between the pucks is $r$r, this mechanism exerts an attractive force of magnitude $Kr$Kr ($K$K is a positive constant) on each puck along the line connecting them. This is mechanically equivalent to being connected by a special spring with natural length $0$0.

At time $t = 0$t=0, puck A is at the origin $(0, 0)$(0,0) and puck B is at point $(0, d)$(0,d) on the $y$y-axis. Simultaneously, puck A is given initial velocity $\vec{v}_A(0) = (v_{Ax}, v_{Ay})$vA​(0)=(vAx​,vAy​) and puck B is given initial velocity $\vec{v}_B(0) = (v_{Bx}, v_{By})$vB​(0)=(vBx​,vBy​).

The two pucks then move under mutual attraction, tracing complex trajectories combining the translational motion of the center of mass and relative motion about the center of mass.

At time $t_1$t1​ ($t > 0$t>0) when the distance between pucks A and B is first maximum, find the kinetic energy $E_K$EK​ of puck B in the stationary coordinate system fixed to the horizontal surface.

Constraints

• Mass of puck A: $m_A = 12 \text{ kg}$mA​=12 kg
• Mass of puck B: $m_B = 4 \text{ kg}$mB​=4 kg
• Proportionality constant of attraction: $K = 75 \text{ N/m}$K=75 N/m
• $y$y-coordinate of puck B at $t = 0$t=0: $d = 6 \text{ m}$d=6 m
• Velocity of puck A at $t = 0$t=0: $\vec{v}_A(0) = (5, 0) \text{ m/s}$vA​(0)=(5,0) m/s
• Velocity of puck B at $t = 0$t=0: $\vec{v}_B(0) = (-20, 10) \text{ m/s}$vB​(0)=(−20,10) m/s

Input Format

The kinetic energy $E_K$EK​ [$\text{J}$J] is expressed as an irreducible fraction $\frac{A}{B}$BA​ with $A, B$A,B coprime positive integers. Give the value of $A + B$A+B as a positive integer.