Problems with moving bodies in physics, when the speed is much less than light, are solved using the laws of Newtonian, or classical mechanics. In it, one of the important concepts is momentum. The basic momentum formulas in physics are given in this article.
Momentum or momentum?
Before we give the momentum formulas of the body in physics, we will get acquainted with this concept. For the first time, a value called impeto (impulse) was used in the description of his works by Galileo at the beginning of the 17th century. Subsequently, Isaac Newton used a different name for her - motus (movement). Since the figure of Newton had a greater influence on the development of classical physics than the personality of Galileo, it was initially customary to speak not about the momentum of the body, but about the amount of movement.
By the amount of motion is understood the product of the speed of the body moving by the inertial coefficient, that is, by the mass. The corresponding formula is:
p¯ = m * v¯
Here p¯ is a vector whose direction coincides with v¯, but the module is m times larger than the module v¯.
Change in p¯
The concept of momentum is currently used less often than momentum. And this fact is connected directly with the laws of Newtonian mechanics. We write it in the form that is given in school textbooks on physics:
F¯ = m * a¯
We replace the acceleration a¯ with the corresponding expression with the derivative of the velocity, we obtain:
F¯ = m * dv¯ / dt
Transferring dt from the denominator of the right-hand side of the equality to the numerator of the left, we obtain:
F¯ * dt = m * dv¯ = dp¯
We obtained an interesting result: in addition to the fact that the acting force F¯ leads to the acceleration of the body (see the first formula of this section), it also changes the amount of its motion. The product of force and time, which is on the left side, is called the impulse of force. It turns out to be equal to the change in p¯. Therefore, the last expression is also called the momentum formula in physics.
Note that dp¯ is also a vector quantity, but, unlike p¯, it is directed not as a velocity v¯, but as a force F¯.
A striking example of a change in the vector of momentum (momentum) is the situation when a soccer player hits the ball. Before hitting the ball moved to the player, after hitting - from him.
The law of conservation of momentum
The formulas in physics that describe the conservation of p¯ can be given in several ways. Before recording them, we answer the question of when the momentum is saved.
We turn to the expression from the previous paragraph:
F¯ * dt = dp¯
It says that if the sum of the external forces affecting the system is zero (closed system, F¯ = 0), then dp¯ = 0, that is, no change in the momentum will occur:
p¯ = const
This expression is common to the momentum of the body and the law of conservation of momentum in physics. We note two important points that you should be aware of in order to successfully apply this expression in practice:
- The impulse is stored along each coordinate, that is, if, prior to some event, the p x value of the system was 2 kg * m / s, then after this event it will be the same.
- The momentum is maintained regardless of the nature of the collisions of solids in the system. Two ideal cases of such collisions are known: absolutely elastic and absolutely plastic impacts. In the first case, kinetic energy is also conserved, in the second part, it is spent on the plastic deformation of bodies, but the momentum is conserved anyway.
Elastic and inelastic interaction of two bodies
A special case of using the momentum formula in physics and its conservation is the movement of two bodies that collide with each other. We consider two fundamentally different cases, which were mentioned in the paragraph above.
If the impact is absolutely elastic, that is, the momentum is transferred from one body to another through elastic deformation, then the conservation formula p is written as follows:
m 1 * v 1 + m 2 * v 2 = m 1 * u 1 + m 2 * u 2
It is important to remember here that the sign of speed should be substituted taking into account its direction along the axis under consideration (opposite speeds have different signs). This formula shows that under the condition of a known initial state of the system (values m 1 , v 1 , m 2 , v 2 ) in the final state (after a collision), there are two unknowns (u 1 , u 2 ). You can find them if you use the corresponding law of conservation of kinetic energy:
m 1 * v 1 2 + m 2 * v 2 2 = m 1 * u 1 2 + m 2 * u 2 2
If the impact is absolutely inelastic or plastic, then after the collision two bodies begin to move as a whole. In this case, the expression
m 1 * v 1 + m 2 * v 2 = (m 1 + m 2 ) * u
As you can see, we are talking about only one unknown (u), therefore, to determine it, this one equality is enough.
Body impulse while moving in a circle
All that was said above about the momentum relates to the linear movements of bodies. What to do in case of rotation of objects around an axis? For this, another concept is introduced in physics, which is similar to a linear momentum. It is called the angular momentum. The formula in physics for him takes the following form:
L¯ = r¯ * p¯
Here r¯ is the vector equal to the distance from the axis of rotation to the particle with momentum p¯, making circular motions around this axis. The quantity L¯ is also a vector, but it is somewhat more difficult to calculate than p¯, since this is a vector product.
The law of conservation L¯
The formula for L¯, which is given above, is the definition of this quantity. In practice, they prefer to use a slightly different expression. We will not go into the details of its receipt (this is not difficult, and everyone can do it on their own), but we will give it right away:
L¯ = I * ω¯
Here I is the moment of inertia (for a material point it is m * r 2 ), which describes the inertial properties of a rotating object, ω¯ is the angular velocity. As you can see, this equation is similar in writing to that for a linear momentum p¯.
If no external forces act on the rotating system (in fact, the moment of forces), then the product of I by ω¯ will be preserved regardless of the processes occurring inside the system. That is, the conservation law for L¯ has the form:
I * ω¯ = const
An example of its manifestation is the performance of athletes in figure skating when they rotate on ice.