Three trains leave their stations 163, 271, and 419 miles apart traveling at 35, 40, and 55 MPH respectively. They're all headed to the same destination station. It's winter so there is icy buildup on two of the three tracks. The first train is coming from a desert location so ice isn't a factor. The question is: How many ping pong balls fit into the number of dentists' electric cars in the Capitol city of Nebraska after the second train reaches halfway?