Dart Thrown at a Cart

Let the positive horizontal direction be defined to the right. The impulse $J$ acting on an object is strictly equal to its change in momentum $\Delta p$. We analyze the momentum change for the dart:

The dart is initially traveling to the right, so $v_{i,\mathrm{dart}} = v_{\mathrm{dart}} > 0$. During the collision, the dart exerts a force on the cart to the right, and by Newton's third law, the cart exerts a force on the dart to the left. This force opposes the motion, causing the dart to slow down or rebound. In either case, the final velocity $v_{f,\mathrm{dart}}$ is algebraically less than the initial velocity $v_{\mathrm{dart}}$:

Substituting this inequality into the impulse equation:

Thus, the dart receives a net negative horizontal impulse. The cart, starting from rest and gaining velocity to the right, receives a positive impulse. The combined system experiences zero net impulse given no external horizontal forces.