# Under What Circumstances is Momentum Conserved?

Momentum, as a conserved quantity, holds great significance in physics. However, there are situations in which momentum is not conserved. So, under what circumstances can we say that momentum is indeed conserved?

## Exploring Conservation of Momentum

To understand when momentum is conserved, we need to consider the scale of the system involved. Even if there are changes in momentum for individual components within the system, it is always possible to find a larger system in which the total momentum remains constant.

Let’s take the example of a football player colliding with a goalpost in the end zone. Upon collision, the player experiences a force that causes them to bounce backward. Interestingly, the Earth also recoils in response to the force applied through the goalpost, thereby conserving momentum. Although the Earth’s recoil is practically negligible due to its immense mass, it is a real effect worth acknowledging.

Now, let’s consider a scenario where two cars collide. Suppose one car (labeled m1) bumps into another car (labeled m2). The force of the collision acts as the only unbalanced force on each car. As a result, car 1 slows down, losing some momentum, while car 2 speeds up, gaining some momentum. Remarkably, the total momentum of the two-car system remains constant throughout this interaction.

Figure 1: A collision between two cars. Car 1 loses momentum while car 2 gains momentum. The total momentum of the system, however, remains constant.

By using the definition of impulse, we can determine the change in momentum for each car involved in the collision. For car 1, the change in momentum (Δp1) is related to the force (F1) acting on it from car 2 for a certain duration of collision (Δt). Similarly, the change in momentum for car 2 (Δp2) is related to the force (F2) acting on it from car 1 during the same duration of collision (Δt). Newton’s third law states that the force exerted by car 1 on car 2 (F2) is equal in magnitude but opposite in direction to the force exerted by car 2 on car 1 (F1).

Consequently, we find that Δp2 = -Δp1. Since the changes in momentum are equal and opposite, the total momentum of the system remains constant:

Δp1 + Δp2 = 0

This outcome emphasizes that the conservation of momentum holds true not only in this particular one-dimensional case, but also for any isolated system with multiple objects. In equation form, the conservation principle is expressed as:

ptot = constant

Here, ptot denotes the total momentum of the system, which is the sum of the momenta of the individual objects. Notably, an isolated system refers to a system with a net external force of zero (Fnet = 0).

## Momentum Conservation in Different Dimensions

It is intriguing to observe that momentum can be conserved differently along different dimensions. For instance, in projectile motion where air resistance is negligible, momentum is conserved in the horizontal direction since there are no horizontal net forces acting on the projectile. In contrast, momentum is not conserved in the vertical direction due to the presence of a non-zero net vertical force. Nevertheless, if we consider the momentum of the entire projectile-Earth system in the vertical direction, the total momentum is conserved.

Figure 2: Conservation of horizontal momentum during projectile motion, while vertical momentum is not conserved.

The conservation of momentum principle applies to a wide array of systems, ranging from a comet colliding with Earth to a gas composed of countless atoms and molecules. The violation of this principle only occurs when there is a non-zero net external force involved. However, we can always expand the system to include the source of the external force, thus conserving momentum. For example, in the collision of two cars mentioned earlier, the two-car system conserves momentum despite each individual car not conserving momentum on its own.

In conclusion, momentum is conserved under specific circumstances within a system. By examining the changes in momentum and the forces exerted, we can determine whether momentum remains constant. The conservation of momentum is a fundamental concept that holds true across various scenarios, helping us better understand the behavior of physical systems and their interactions. To explore more scientific topics, visit 5 WS.

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