Momentum and its conservation

Two carts collide, stick together and roll away as one. The impact force changed rapidly, and we did not measure it. Can we still find their common velocity? Let us look for a quantity that the carts exchange during their interaction.

A heavy cart is harder to stop than a light cart moving at the same speed; the same cart is harder to stop when it moves faster. We need a quantity that accounts for both factors. In classical mechanics, the momentum of a body is \vec p=m\vec v, measured in kg·m/s. Doubling either mass or velocity doubles momentum when the other factor is unchanged. At rest, momentum is zero. Its meaning will become clearer when we examine how a force changes it.

Two identical carts approach each other at equal speeds. Each has nonzero momentum, but their vector sum is zero because the directions are opposite. If they move in the same direction, the sum is twice the momentum of one cart. So the momentum of a system cannot be found just by adding magnitudes. A zero total does not mean that nothing moves inside the system: it expresses a balance of directed momenta.

A ball hits a wall and rebounds at its original speed. Has its momentum changed? Yes: if p_x=mv before the collision and p'_x=-mv afterwards, then \Delta p_x=-2mv. Simply stopping it would produce half as much change. Here \Delta\vec p=\vec p'-\vec p, the contact time is \Delta t in seconds, and the average net force is \vec F_{\text{ср}} in newtons. Vector subtraction retains the information about the reversal.

For constant mass, Newton's law relates force to the change in velocity, and hence in momentum. During contact, \vec F_{\text{ср}}\Delta t=\Delta\vec p; this product is called impulse. Thus \vec F_{\text{ср}}=\Delta\vec p/\Delta t. Doubling the time for the same stop halves the average force: this is how a soft cushion helps. This conclusion concerns the average force, not its maximum; an average alone does not determine the shape of a brief impact peak.

Take both carts as one system. During the collision, their mutual forces are equal and opposite, act for the same time and produce opposite changes in momentum. These cancel when added. External forces, such as friction with the track, can change the total momentum. If their net impulse during the short collision is small, total momentum is approximately conserved. The horizontal component can also be conserved when balanced vertical forces are present.

To keep the instants separate, draw two states of the same system: immediately before and after the collision. The masses m_1,m_2 are unchanged; denote velocities before the interaction by \vec v_1,\vec v_2 and afterwards by \vec v'_1,\vec v'_2. A prime marks the new state, and all velocities are measured in the same chosen reference frame. If the bodies stick together, they have one final velocity; without sticking, we cannot assume their velocities are equal.

The balance is now m_1\vec v_1+m_2\vec v_2=m_1\vec v'_1+m_2\vec v'_2 when the external impulse is negligible. In one dimension, use signed components. When a moving body sticks to a stationary one, v'=m_1v_1/(m_1+m_2): the same total momentum is carried by a larger mass, so the speed is lower. If the second body is very massive, their common speed is close to zero. By itself, this equation says nothing about conservation of mechanical energy.

A 2 kg cart moves to the right at 3 m/s and sticks to a stationary 1 kg cart. Neglect the external horizontal impulse. Find their common velocity.

The system consists of both carts, with right chosen as positive. The external impulse is small, so total momentum is conserved.

Before impact, p=2×3+1×0=6 кг·м/с.

After they stick together, the mass is 3 kg and the velocity is shared: 3v=6, so v=2 м/с.

Check: both carts move in the original direction, but more slowly than the first cart did; the momentum 3×2 is again 6.

Their common velocity is 2 m/s to the right. The total momentum of 6 kg·m/s now belongs to 3 kg instead of the 2 kg that was moving before, so the common speed is below the initial 3 m/s.

Momentum combines mass with the magnitude and direction of velocity; add its vectors, not just positive magnitudes.

Internal interactions redistribute momentum among bodies; external impulse changes the total.

For the same momentum change, longer contact means a smaller average force; momentum conservation does not imply conservation of mechanical energy.

Next, we will distinguish momentum conservation from mechanical energy conservation.