Often in physics they talk about the momentum of the body, implying the amount of movement. In fact, this concept is closely related to a completely different quantity - with force. The impulse of power is what it is, how it is introduced into physics, and what is its meaning: all these questions are covered in detail in the article.
Amount of movement
The impulse of the body and the impulse of force are two interconnected quantities; moreover, they practically mean the same thing. First, we analyze the concept of momentum.
The quantity of movement 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: this is Galileo Galilei, the famous Italian, who called the discussed value impeto (impulse), and Isaac Newton, the great Englishman, who in addition to the magnitude of motus (movement) also used the concept of vis motrix (driving force).
So, the aforementioned scientists by the amount of movement understood the product of the mass of the object by the speed of its linear movement in space. This definition in the language of mathematics is written as follows:
p¯ = m * v¯
Note that we are talking about the vector (p¯) directed towards the body motion, which is proportional to the velocity modulus, and the mass of the body plays the role of the proportionality coefficient.
The relation between the momentum of force and the change in p¯
As mentioned above, in addition to the amount of motion, Newton also introduced the concept of driving force. He defined this value as follows:
F¯ = m * a¯
This is the familiar law of the appearance of the acceleration a¯ of a body as a result of the action of some external force F¯ on it. This important formula allows us to derive the law of momentum of force. Note that a¯ is the time derivative of the velocity (the rate of change of v¯), which means the following:
F¯ = m * dv¯ / dt or F¯ * dt = m * dv¯ =>
F¯ * dt = dp¯, where dp¯ = m * dv¯
The first formula in the second line is the momentum of the force, that is, the value equal to the product of the force for the period of time during which it acts on the body. It is measured in Newtons per second.
Formula Analysis
The expression for the momentum of force in the previous paragraph also reveals the physical meaning of this quantity: it shows how much the momentum dt changes during the time interval. Note that this change (dp¯) is completely independent of the total momentum of the body. The force impulse is the reason for the change in the momentum, which can lead both to an increase in the latter (when the angle between the force F¯ and the speed v¯ is less than 90 o ), and to its decrease (the angle between F¯ and v¯ is more than 90 o ).
An important conclusion follows from the analysis of the formula: the units of measurement of the force momentum coincide with those for p¯ (newton per second and kilogram per meter per second), moreover, the first quantity is equal to the change in the second, therefore, instead of the force momentum, the phrase "body momentum" is often used, although it’s more correct to say “change in momentum”.
Time-independent and time-dependent forces
Above, the law of momentum of force was presented in differential form. To calculate the value of this quantity, it is necessary to integrate over the time of action. Then we get the formula:
∫ t1 t2 F¯ (t) * dt = Δp¯
Here, the force F¯ (t) acts on the body for the time Δt = t2-t1, which leads to a change in the momentum by Δp¯. As you can see, the momentum of a force is a quantity determined by a force that depends on time.
Now we will 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¯ * ∫ t1 t2 dt = Δp¯ => F¯ * (t2-t1) = Δp¯
The last equality allows us to calculate the momentum of constant force.
When solving real problems of changing the momentum, despite the fact that the force in the general case depends on the time of action, it is assumed to be constant and some effective average value F¯ is calculated.
Examples of the manifestation of the impulse of force
What role this quantity plays is easiest to understand with concrete examples from practice. Before we bring them, we write out the corresponding formula again:
F¯ * Δt = Δp¯
Note that if Δp¯ is a constant, then the modulus of the force momentum is also a constant, so the larger Δt, the smaller F¯, and vice versa.
Now we give specific examples of the impulse of force 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 earth's surface (support reaction force F¯), thereby reducing its strength.
- The boxer, deflecting his head from the blow, extends the contact time Δt of the opponent’s gloves with his face, reducing the impact force.
- Modern cars are trying to design in such a way that in case of a collision their body deforms as much as possible (deformation is a process that develops over time, which leads to a significant decrease in collision force and, as a result, a decrease in the risks of passenger damage).
The concept of the moment of force and its momentum
The moment of force and momentum of this moment are other quantities that are different from those considered above, since they concern not rotational, but linear 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 influence of the force) and the force itself, that is, the formula is true:
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 take the wrench away from the nut (large 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 period of time during which it acts on the rotating system, that is:
M¯ * Δt = ΔL¯
The quantity ΔL¯ is called the change in angular momentum, or angular momentum. The last equation is important for considering systems with a rotation axis, because it shows that the angular momentum of the system will be preserved if there are no external forces that create the moment M¯, which is mathematically written 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.
The task of a collision of a bird and an airplane
This problem is not fantastic. Such clashes do occur quite often. Thus, according to some reports, in 1972, in the airspace of Israel (the zone of the most dense bird migration), about 2.5 thousand bird collisions with combat and transport aircraft, as well as with helicopters, were recorded.
The task is as follows: it is necessary to approximately calculate what impact force falls on the bird if an airplane flying at a speed of v = 800 km / h is encountered in its path of movement.
Before proceeding with the decision, we assume that the length of the bird in flight is l = 0.5 meters, and its weight is m = 4 kg (this can be, for example, a drake or a goose).
We neglect the speed of the bird (it is small in comparison with that for the plane), and we will also consider the mass of the plane to be much larger than the bird. These approximations allow us to say that the change in the momentum of the bird is equal to:
Δp = m * v
To calculate the impact force F, it is necessary to know the duration of this incident, it is approximately equal to:
Δt = l / v
Combining these two formulas, we obtain the desired expression:
F = Δp / Δt = m * v 2 / l.
Substituting the figures from the conditions of the problem into it, we obtain F = 395062 N.
It will be more obvious to translate this figure into equivalent mass using the formula for body weight. Then we get: F = 395062 / 9.81 ≈ 40 tons! In other words, the bird perceives a collision with the plane as if 40 tons of cargo fell on it.