Linear momentum and its conservation — common mistakes

Remember that $\Delta$ (delta) signifies 'change in'. So, $\Delta p$ means final momentum minus initial momentum ($p_f - p_i$). This formula highlights that a larger force causes a faster change in momentum, and a force applied for a longer time causes a larger change in momentum.

A common misconception is that kinetic energy is always conserved if momentum is. Remember, only in perfectly elastic collisions is kinetic energy conserved. In all inelastic collisions, momentum is conserved, but kinetic energy is not.

For a two-body explosion from rest, $m_1v_1 + m_2v_2 = 0$, which means $m_1v_1 = -m_2v_2$. The negative sign indicates that the velocities are in opposite directions. This is the principle behind rocket propulsion and the recoil of a gun.

Linear momentum is a vector quantity representing an object's 'quantity of motion', defined as the product of its mass and velocity ($p = mv$). It's crucial because it allows us to analyse interactions like collisions and explosions, especially through the Principle of Conservation of Momentum, which predicts outcomes when forces are internal to a system.

Linear momentum is conserved in a closed system, meaning no mass or energy enters or leaves. Crucially, it is conserved only when the net external force acting on the system is zero. Internal forces (like the forces of collision between objects) do not change the total momentum of the system.

You can distinguish between elastic and inelastic collisions by examining the total kinetic energy of the system. In an elastic collision, the total kinetic energy before the impact is equal to the total kinetic energy after. In an inelastic collision, the total kinetic energy is not conserved; some of it is transformed into other forms like heat, sound, or energy used for deformation, making the final kinetic energy less than the initial.

Force is the rate of change of momentum ($F = \Delta p / \Delta t$). Impulse is the total change in momentum ($J = \Delta p$). Therefore, impulse is also equal to the average force multiplied by the time it acts ($J = F \Delta t$). A large impulse can be delivered by a large force over a short time, or a smaller force over a long time.