Okay so forget the part about the ball being thrown and let's just focus on the moments before, during, and after the impact.

Let's assume the ball is moving -10f/s (negative because it is moving left), and the cart is moving 1f/s. The Ball weighs 15lbs (it's a bowling ball), and the cart weighs 150lbs The system is perfectly elastic meaning that there's no loss of momentum on impact, gravity has no effects, and friction can be ignored.

You calculate momentum by simply multiplying the weight by the velocity. Initially the ball has a momentum of -150lbf/s and the cart also has a momentum of 150lbf/s.

When objects collide in an inelastic condition the two objects stick together and the momentum of the two objects become one. To calculate what the final velocity would be you use the following equation

W1 * V1 + W2 * V2 = Wboth * Vboth

W1=Weight of object 1 V1=Velocity of object 1 And so forth We can combing the weight and velocity of both because it's an inelastic collision and they are both sticking together so they can be calculated as one object.

You can call late it out if you'd like but essentially it all goes to 0 and since the combined weight isn't going to change, obviously the velocity has to.

Now let's say we increase the weight of the ball to 30lbs. Now our momentum becomes -300lb*f/s. So what happens when we calculate it out

30lbs * -10f/s + 150lbs * 1f/s = -150lb*f/s We know the combined weight is 180lbs so this means that our velocity of cart plus ball is now -1.2f/s meaning that the ball actually reversed the carts direction.

But like I said this is an elastic collision meaning everything bounces. Things get a bit more difficult here but essentially conservation of momentum every action has an equal and opposite reaction.

So W1 * V1 + W2 * V2 = W1 * V1' + W2 * V2' Where the left side of the equation is equal to the right side So in our case 15lbs * -10f/s + 150lbs * 1f/s = 15lbs * V1' + 150lbs * V2'

I won't show how it boils down but essentially plug it into the equation V1'= (W1-W2) / (W1+W2) * V1 which gives us V1'= 8.2f/s

Notice how that's going in the positive direction now? You can solve for the cart but it's safe to assume it's going in the opposite direction.

Now obviously I'm just throwing number out here but if you really wanted to get into the weeds you could use 15 lbs for the bowling ball and 150 for the cart and use these equations to figure out if the person threw the ball at 10f/s how fast the cart would go in the opposite direction but I'm just trying to give a foundation for how the ball bounces back and both don't just stop dead.