We hear the word momentum a lot, especially in sports. When a team, for instance, is playing well and winning games consistently, we say that team has momentum. While we may not use this word in a quantitative sense in everyday life, momentum is actually related to mathematics when it comes to physics.
What is linear momentum?
Any object with mass that is moving has momentum. In this explanation, we look at objects moving linearly. So what is linear momentum?
Linear momentum is the product of the mass and velocity of an object.
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.
Impulse is the change of momentum of an object when a force is applied over a certain duration of time.
\[Impulse = F \Delta t\]
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?
The impulse-momentum theorem simply states that the change in impulse is equal to the change in momentum.
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, mvf is the final momentum and mvi 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.
Conservation of linear momentum: The total momentum in an isolated system where no external forces occur is conserved. The total momentum before the collision between two objects will be equal to the total momentum after the collision. The total energy is also conserved for such a system.
Suppose you have two objects of masses m1 and m2 heading towards each other with velocities u1 and u2.
Two objects about to collide
Both objects collide with each other after some time and exert forces F1 and F2 on each other.
When the objects collide, they exert a force on each other, causing them to stop for an instant
After the collision, the two objects will move in the opposite direction with velocities v1 and v2 respectively.
After the collision, both objects move in the opposite direction with different velocities, Usama Adeel – StudySmarter Originals
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
\[m_1v_1 - m_1u_1 = -m_2v_2 + m_2u_2\]
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
A collision happens whenever an object in motion comes into contact with another object that is at rest or in motion.
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.
Now, you might be wondering, whenever someone kicks a ball, the foot of the person doesn’t go off in a separate direction (that would be terrible if it did!). So, what kind of a collision is this?
Many collisions are not perfectly elastic, like a soccer player kicking a ball for instance. But, the foot of the player and the ball do remain separate after the player kicks the ball. Before a player kicks the ball, the ball is at rest and the foot moves with a high velocity. After the player kicks the ball, the ball goes in the direction in which it is kicked.
We refer to all these scenarios as nearly elastic collisions because some form of energy is converted to sound or heat, etc.
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 vf will depend on the magnitudes and directions of the two initial velocities.
Sometimes, we can approximate car crashes as perfectly inelastic collisions where the total momentum is conserved. However, the total energy is not conserved because some energy is converted into sound, heat, and internal energy. The crashed cars will never return to their original position after the collision, which is why these types of collisions are named inelastic.
- With elastic collisions, the total momentum and total energy are conserved.
- With inelastic collisions, the total momentum is conserved but the total energy is not conserved.
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.
Momentum - Key takeaways
- Momentum is the product of the mass and velocity of an object.
- The higher the momentum, the more force will be required to stop an object.
- Impulse is the force applied over a certain interval of time.
- The impulse-momentum theorem states that impulse is the change in momentum. The area under a force-time graph gives the impulse.
- The law of conservation of momentum states that the total momentum before the collision between two objects is the same as the total momentum after the collision.
- With elastic collisions, the total momentum and total energy are conserved.
- With inelastic collisions, the total momentum is conserved but the total energy is not conserved.
ImagesAn elastic collision between two particles with the same mass, one of which is at rest. https://commons.wikimedia.org/wiki/File:Elastic_collision.svg
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