Conservation of Energy/Linear&Angular Momentum

If a ball hits a rod at rest at any position along the rod, the rod will be moving with the same linear velocity in each case. However, if the ball hits the rod away from its center of mass, the rod will also rotate.

How does this not violate the conservation of energy? The incoming ball has the same mass and velocity in each case and thus the same energy is inputted to the system, and the rod will have the same mass and linear velocity in each case, plus excess rotational energy. How can simply choosing where the ball hits the rod add energy to the system? My best guess is that hitting the rod off-center somehow "drains" more energy from the ball and slows it down more after collision, but I dont quite see why this would occur mechanistically.

• Why would it have same velocity in all cases? – Sidharth Giri Apr 18 at 16:35
• I'm assuming thats the case because linear momentum has to be conserved, and I've seen someone corroborate my thought elsewhere on the internet. – jizzle nizzle Apr 18 at 16:39
• In reasoning that the linear velocity of the rod is the same in all cases, you are making the false implicit assumption that the rebound velocity of the ball is the same in all cases. – Ben51 Apr 18 at 17:03
• @Ben51 Ah, the ball would feel less resistance the further out from the center of mass it hits, so it would rebound less. And if it rebounds less, that means its kept less of its kinetic energy and poured more into the rod, explaining how the rod gets excess energy to rotate with. – jizzle nizzle Apr 18 at 17:15
• Also, the linear motion of the rod is slower following a collision near the end than one near the middle. So not only in the KE of the rebounding ball less after an off center collision, the translational KE of the rod is also less. For elastic collision, the sum of these two and the rotational KE of the rod is always equal to the initial KE of the ball. – Ben51 Apr 18 at 17:20