Often in physics they talk about the momentum of a body, implying the amount of motion. In fact, this concept is closely connected with a completely different quantity - with force. The impulse of force - what is it, how is it introduced into physics, and what is its meaning: all these issues are covered in detail in the article.
Amount of movement
The momentum of the body and the momentum of the force are two interrelated quantities, moreover, they practically mean the same thing. First, let's analyze the concept of momentum.
The amount of motion as a physical quantity first appeared in the scientific works of modern scientists, in particular in the 17th century. It is important to note two figures here: Galileo Galilei, the famous Italian, who called the quantity under discussion impeto (momentum), and Isaac Newton, the great Englishman, who, in addition to the motus (motion) quantity, also used the concept of vis motrix (driving force).
So, the named scientists under the amount of motion understood the product of the mass of an object and the speed of its linear movement in space. This definition in the language of mathematics is written as follows:
p¯=mv¯
Note that we are talking about the vector value (p¯), directed in the direction of body movement, which is proportional to the speed modulus, and the body mass plays the role of the proportionality coefficient.
Relationship between the momentum of force and the change in p¯
As mentioned above, in addition to the momentum, Newton also introduced the concept of driving force. He defined this value as follows:
F¯=ma¯
This is the familiar law of the appearance of acceleration a¯ on a body as a result of some external force F¯ acting on it. This important formula allows us to derive the law of momentum of force. Note that a¯ is the time derivative of the rate (the rate of change of v¯), which means:
F¯=mdv¯/dt or F¯dt=mdv¯=>
F¯dt=dp¯, where dp¯=mdv¯
The first formula in the second line is the impulse of the force, that is, the value equal to the product of the force and the time interval during which it acts on the body. It is measured in newtons per second.
Formula analysis
The expression for the impulse of force in the previous paragraph also reveals the physical meaning of this quantity: it shows how much the momentum changes over a period of time dt. Note that this change (dp¯) is completely independent of the total momentum of the body. The impulse of a force is the cause of a change in momentum, which can lead to bothan increase in the latter (when the angle between the force F¯ and speed v¯ is less than 90o), and to its decrease (the angle between F¯ and v¯ is greater than 90o).
From the analysis of the formula, an important conclusion follows: the units of measurement of the impulse of force are the same as those for p¯ (newton per second and kilogram per meter per second), moreover, the first value is equal to the change in the second, therefore, instead of the impulse of force, the phrase is often used "momentum of the body", although it is more correct to say "change in momentum".
Forces dependent and independent of time
The force impulse law was presented above in differential form. To calculate the value of this quantity, it is necessary to integrate over the action time. Then we get the formula:
∫t1t2 F¯(t)dt=Δp¯
Here, the force F¯(t) acts on the body during the time Δt=t2-t1, which leads to a change in momentum by Δp¯. As you can see, the momentum of a force is a quantity determined by a time-dependent force.
Now consider a simpler situation, which is realized in a number of experimental cases: we assume that the force does not depend on time, then we can easily take the integral and get a simple formula:
F¯∫t1t2 dt=Δp¯ =>F¯(t2-t1)=Δp¯
The last equation allows you to calculate the momentum of a constant force.
When decidingreal problems on changing the momentum, despite the fact that the force generally depends on the action time, it is assumed to be constant and some effective average value F¯ is calculated.
Examples of manifestation in practice of an impulse of force
What role does this value play, it is easiest to understand on specific examples from practice. Before giving them, let's write out the corresponding formula again:
F¯Δt=Δp¯
Note, if Δp¯ is a constant value, then the momentum modulus of the force is also a constant, so the larger Δt, the smaller F¯, and vice versa.
Now let's give concrete examples of momentum in action:
- A person who jumps from any height to the ground tries to bend his knees when landing, thereby increasing the time Δt of the impact of the ground surface (support reaction force F¯), thereby reducing its strength.
- The boxer, by deflecting his head from the blow, prolongs the contact time Δt of the opponent's glove with his face, reducing the impact force.
- Modern cars are trying to be designed in such a way that in the event of a collision, their body is deformed as much as possible (deformation is a process that develops over time, which leads to a significant decrease in the force of a collision and, as a result, a decrease in the risk of injury to passengers).
The concept of the moment of force and its momentum
Moment of force and momentumthis moment, these are other quantities different from those considered above, since they no longer relate to linear, but to rotational motion. So, the moment of force M¯ is defined as the vector product of the shoulder (the distance from the axis of rotation to the point of action of the force) and the force itself, that is, the formula is valid:
M¯=d¯F¯
The moment of force reflects the ability of the latter to perform torsion of the system around the axis. For example, if you hold the wrench away from the nut (big lever d¯), you can create a large moment M¯, which will allow you to unscrew the nut.
By analogy with the linear case, the momentum M¯ can be obtained by multiplying it by the time interval during which it acts on a rotating system, that is:
M¯Δt=ΔL¯
The value ΔL¯ is called the change in angular momentum, or angular momentum. The last equation is important for considering systems with an axis of rotation, because it shows that the angular momentum of the system will be conserved if there are no external forces that create the moment M¯, which is written mathematically as follows:
If M¯=0 then L¯=const
Thus, both momentum equations (for linear and circular motion) turn out to be similar in terms of their physical meaning and mathematical consequences.
Bird-Aircraft Collision Problem
This problem is not something fantastic. These collisions do happen.often. Thus, according to some data, in 1972, about 2.5 thousand bird collisions with combat and transport aircraft, as well as with helicopters, were recorded in Israeli airspace (the zone of the densest bird migration)
The task is as follows: it is necessary to approximately calculate how much impact force falls on a bird if an airplane flying at a speed of v=800 km/h is encountered on its path.
Before proceeding to the decision, let's assume that the length of the bird in flight is l=0.5 meters, and its mass is m=4 kg (it can be, for example, a drake or a goose).
Let's neglect the speed of the bird (it is small compared to that of the aircraft), and we will also consider the mass of the aircraft to be much greater than that of the birds. These approximations allow us to say that the change in the momentum of the bird is:
Δp=mv
To calculate the impact force F, you need to know the duration of this incident, it is approximately equal to:
Δt=l/v
Combining these two formulas, we get the required expression:
F=Δp/Δt=mv2/l.
Substituting the numbers from the condition of the problem into it, we get F=395062 N.
It will be more visual to translate this figure into an equivalent mass using the formula for body weight. Then we get: F=395062/9.81 ≈ 40 tons! In other words, a bird perceives a collision with an airplane as if 40 tons of cargo had fallen on it.
