Momentum

Calculating momentum

The momentum of any object depends on two things: mass and velocity. We can express it mathematically as:

\[p = m \cdot v\]

Here, p is the momentum, m is the mass measured in kilograms (kg), and v is the velocity measured in metres per second (m/s). Momentum is a vector quantity with units of kg⋅m/s. As we can see from the equation, an object’s momentum will increase if its velocity increases (directly proportional relationship). The more momentum an object has, the more force it needs to stop.

What is the impulse?

Suppose you are driving a car that has a certain momentum. That momentum will depend on the mass of the car and the velocity at which it is moving. Now, let’s say that you want to bring the car to a stop. How would you do it?

First, you will slam on the brakes, which will quickly bring the car to rest via the large deceleration force applied. The deceleration force it takes to stop the car depends on the momentum of the car.

Another way to bring the car to rest is to take the foot off of the pedal and let friction come into play. In this scenario, a small amount of force is applied over a long duration of time.

Either way, the moving car will come to a rest, but what is this force that is required to bring a moving body to rest? This is called the impulse.

The units of impulse are Newton seconds (N·s). As a result, the area under a force-time graph will yield the impulse or change in momentum.

What is the impulse-momentum theorem?

We express this mathematically as follows:

\[F \Delta t = \Delta p\]

If we further break down the change in momentum, we get:

\[F \Delta t = mv_f - mv_i\]

Here, mv_f is the final momentum and mv_i is the initial momentum.

The rate of change of momentum can be expressed as:

\[F = \frac{m(v-u)}{\Delta t}\]

Here, v is the final velocity and u is the initial velocity.

What is the law of conservation of linear momentum?

Just like in chemistry, we have the law of conservation of matter, and in physics, we have the law of conservation of energy. We can extend these concepts to form another law known as the law of conservation of momentum.

Suppose you have two objects of masses m₁ and m₂ heading towards each other with velocities u₁ and u₂.

Both objects collide with each other after some time and exert forces F₁ and F₂ on each other.

After the collision, the two objects will move in the opposite direction with velocities v₁ and v₂ respectively.

As the law of conservation of linear momentum states that the momentum of the colliding objects is conserved, we can derive the following equation:

\[F_1 = -F_2\]

\[\frac{m_1(v_1-u_1)}{t_1} = - \frac{m_2(v_2 - u_2)}{t_2}\]

Since t₁ and t₂ are the same because both objects collided for the same amount of time, we can reduce the equation to

Rearranging the above yields

\[m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\]

This equation states the conservation of linear momentum (i.e the total momentum before the collision is equal to the total momentum after the collision). After impact, the velocities change but the masses remain constant.

Momentum: are there different types of collisions?

Not every collision results in objects moving apart separately. There are scenarios, for example, where the objects collide and sometimes combine, forming new objects. Keep in mind that the linear momentum is conserved in any type of collision.

Collisions

Elastic collisions

With elastic collisions, the objects that come into contact remain separate. In other words, the objects don’t combine to form a new object. The total kinetic energy and momentum are conserved in this type of collision, which is why the objects bounce off one another without the loss of any energy.

Perfectly inelastic collisions

In these types of collisions, the objects collide and move together as one mass. When we examine perfectly inelastic collisions, we can treat the two separate objects as a single object after the collision. Hence, in terms of momentum:

\[p_1 + p_2 = p_{total}\]

\[m_1v_1 + m_2+v_2 = (m_1+m_2)v_f\]

Note that v_f will depend on the magnitudes and directions of the two initial velocities.

Conclusion of collisions

In real life, no collision is elastic or perfectly inelastic but is somewhere in between, which we can simply label as inelastic collisions because they imply that some energy is lost as a result of the collisions. However, we often approximate a collision to either of the extremes to make the calculations simpler.

An elastic collision between two particles with the same mass, one of which is at rest. https://commons.wikimedia.org/wiki/File:Elastic_collision.svg