I'm sorry where did you see in the first principle of thermodynamics that you should only count energy moving in some direction?
I can't see it ![Confused :-//](https://www.eevblog.com/forum/Smileys/default/confused0024.gif)
So let's say your ball has a mass of one ton, moving at 10 m/s. There is a collision with a bat, with 2J of energy (your number).
Compute the recoil, i.e. the momentum change of the bat.
I do not think I asked you about aphantasia.
Can you visualize the difference between when the ball moves in the direction of travel and when it moves in the opposite direction ?
If you are in that vehicle looking out at balls traveling at 10m/s
when vehicle is at 5m/s you will see balls coming from the back of your vehicle and moving past your vehicle at around 5m/s so all you need to do is still the bat out and the ball will hit it providing you with energy to increase vehicle speed.
When vehicle is at 15m/s then you see balls moving in the opposite direction so come from the front of your vehicle and move backward so if you stick out the bat and ball hits it it will slow your vehicle down.
You will need to prepare to swing that bat at high speed to heat that vehicle after it just passed the vehicle and so you need quite some energy to hit that ball in order to accelerate your vehicle. So much energy that even at 100% efficiency it will be barely equal with the amount of energy you lost by braking the vehicle.
Hope you can see the difference between a ball coming from the back of the vehicle at 5m/s and one coming from the front of vehicle at 5m/s
They both have equal speed and mass but the direction of travel is different thus one is positive and the other is negative. One can accelerate the vehicle and the other can slow down the vehicle by the same amount.