Vector quantity in physics. Examples of vector quantities

Physics and mathematics cannot do without the concept of “vector quantity”. It is necessary to know and recognize it, as well as to be able to operate with it. This is definitely worth learning to avoid confusion and stupid mistakes.

How to distinguish scalar from vector?

The first always has only one characteristic. This is its numerical value. Most scalar values ​​can take both positive and negative values. Examples include electrical charge, work, or temperature. But there are scalars that cannot be negative, for example, length and mass.

The vector quantity, in addition to the numerical value, which is always taken modulo, is also characterized by direction. Therefore, it can be depicted graphically, that is, in the form of an arrow, the length of which is equal to the absolute value of the quantity directed in a certain direction.

When writing, each vector value is indicated by an arrow on the letter. If we are talking about a numerical value, then the arrow is not written or it is taken modulo.

What actions are most often performed with vectors?

First up is a comparison. They can be equal or not. In the first case, their modules are the same. But this is not the only condition. They should still have the same or opposite directions. In the first case, they should be called equal vectors. In the second, they turn out to be opposite. If at least one of the indicated conditions is not satisfied, then the vectors are not equal.

Then comes the addition. It can be done according to two rules: a triangle or a parallelogram. The first prescribes to postpone first one vector, then from the end of the second. The result of the addition will be the one that needs to be drawn from the beginning of the first to the end of the second.

The parallelogram rule can be used when you need to add vector quantities in physics. Unlike the first rule, here they should be put off from one point. Then finish them up to a parallelogram. The result of the action should be considered the diagonal of a parallelogram drawn from the same point.

If the vector value is subtracted from another, then they are again delayed from one point. Only the result will be a vector that matches the one that is delayed from the end of the second to the end of the first.

What vectors are studied in physics?

There are as many as scalars. You can simply remember what vector quantities exist in physics. Or know the signs by which they can be calculated. Those who prefer the first option will need such a table. It lists the basic vector physical quantities.

Now a little more about some of these values.

The first value is speed

It is worth starting to give examples of vector quantities. This is due to the fact that it is studied among the first.

Speed ​​is defined as a characteristic of the motion of a body in space. It sets the numerical value and direction. Therefore, speed is a vector quantity. In addition, it is customary to divide it into species. The first is linear speed. It is introduced when considering rectilinear uniform motion. Moreover, it turns out to be equal to the ratio of the path traveled by the body to the time of movement.

The same formula can be used for uneven movement. Only then will it be average. Moreover, the time interval that must be selected must be as small as possible. When the time interval tends to zero, the speed value is already instantaneous.

If arbitrary motion is considered, then here always speed is a vector quantity. After all, it has to be decomposed into components directed along each vector directing the coordinate lines. Moreover, it is defined as the derivative of the radius vector, taken over time.

The second value is force.

It determines a measure of the intensity of the impact that is exerted on the body by other bodies or fields. Since force is a vector quantity, then it necessarily has its value in absolute value and direction. Since it acts on the body, the point to which the force is applied is also important. To get a visual representation of the force vectors, you can refer to the following table.

Also a vector quantity is the resultant force. It is defined as the sum of all mechanical forces acting on the body. To determine it, it is necessary to perform addition according to the principle of the triangle rule. Only postpone vectors need to take turns from the end of the previous one. The result will be one that connects the beginning of the first with the end of the last.

The third value is displacement.

During movement, the body describes a certain line. It is called a trajectory. This line can be completely different. More important is not her appearance, but the points of the beginning and end of the movement. They are connected by a segment called movement. This is also a vector quantity. Moreover, it is always directed from the beginning of the movement to the point where the movement was stopped. It is accepted to designate it with the Latin letter r.

Here the following question may appear: "Is the path a vector quantity?" In the general case, this statement is not true. The path is equal to the length of the path and does not have a specific direction. An exception is the situation when rectilinear motion in one direction is considered. Then the module of the displacement vector coincides in value with the path, and their direction is the same. Therefore, when considering movement along a straight line without changing the direction of movement, the path can be included in examples of vector quantities.

Fourth Value - Acceleration

It is a characteristic of the speed of a change in speed. Moreover, acceleration can have both positive and negative values. In a rectilinear motion, it is directed towards higher speed. If the movement occurs along a curved path, then its acceleration vector is decomposed into two components, one of which is directed to the center of curvature along the radius.

The average and instantaneous acceleration values ​​are distinguished. The first should be calculated as the ratio of the change in speed over a certain period of time to this time. When the time interval tends to zero, they speak of instantaneous acceleration.

The fifth value is momentum

In another way, it is also called the amount of movement. Momentum is a vector quantity due to the fact that it is directly related to the speed and force applied to the body. Both of them have a direction and set its impulse.

By definition, the latter is equal to the product of body mass and speed. Using the concept of momentum of a body, one can write down the famous Newton's law in a different way. It turns out that the change in momentum is equal to the product of force by the time interval.

An important role in physics is played by the law of conservation of momentum, which states that in a closed system of bodies its total momentum is constant.

We very briefly listed which quantities (vector) are studied in a physics course.

Inelastic Impact Problem

Condition. There is a fixed platform on the rails. A car is approaching it at a speed of 4 m / s. The masses of the platform and the car are 10 and 40 tons, respectively. The wagon hits the platform, auto-coupling occurs. It is necessary to calculate the speed of the platform-car system after an impact.

Decision. First, you need to enter the notation: the speed of the car before the impact is v ₁ , the car with the platform after coupling is v, the mass of the car is m ₁ , the platform is m ₂ . By the condition of the problem, it is necessary to find out the value of speed v.

The rules for solving such tasks require a schematic representation of the system before and after interaction. It is reasonable to direct the OX axis along the rails in the direction where the car moves.

Under these conditions, the wagon system can be considered closed. This is determined by the fact that external forces can be neglected. The force of gravity and the reaction of the support are balanced, and friction on the rails is not taken into account.

According to the law of conservation of momentum, their vector sum before the interaction of the car and the platform is equal to the total for coupling after impact. At first, the platform did not move, so its momentum was zero. Only the wagon moved, its momentum is the product of m ₁ and v ₁ .

Since the impact was inelastic, that is, the wagon adhered to the platform, and then they began to roll together in the same direction, the momentum of the system did not change direction. But its meaning has become different. Namely, the product of the sum of the mass of the wagon with the platform and the desired speed.

We can write the following equality: m ₁ * v ₁ = (m ₁ + m ₂ ) * v. It will be true for the projection of the pulse vectors onto the selected axis. It is easy to deduce from it the equality that is required to calculate the desired speed: v = m ₁ * v ₁ / (m ₁ + m ₂ ).

According to the rules, the values ​​for the mass should be converted from tons to kilograms. Therefore, when substituting them in the formula, you must first multiply the known values ​​by a thousand. Simple calculations give the number 0.75 m / s.

Answer. The speed of the wagon with the platform is 0.75 m / s.

The task of dividing the body into parts

Condition . The speed of a flying grenade is 20 m / s. It is torn into two fragments. The mass of the first is 1.8 kg. He continues to move in the direction in which the grenade flew at a speed of 50 m / s. The second fragment has a mass of 1.2 kg. What is its speed?

Decision. Let the masses of fragments are denoted by the letters m ₁ and m ₂ . Their speeds will be v ₁ and v _2, respectively. The initial speed of the grenade is v. In the problem, you need to calculate the value of v ₂ .

In order for the larger fragment to continue to move in the same direction as the whole grenade, the second must fly in the opposite direction. If we choose for the direction of the axis that which was at the initial impulse, then after breaking, a large fragment flies along the axis, and a small one - against the axis.

In this problem, it is allowed to use the law of conservation of momentum due to the fact that the grenade rupture occurs instantly. Therefore, despite the fact that the grenade and its parts are affected by gravity, it does not have time to act and change the direction of the pulse vector with its value modulo.

The sum of the vector values ​​of the momentum after the grenade rupture is equal to that which came before it. If we write down the law of conservation of the momentum of the body in the projection onto the OX axis, then it will look like this: (m ₁ + m ₂ ) * v = m ₁ * v ₁ - m ₂ * v ₂ . From it, simply express the desired speed. It will be determined by the formula: v ₂ = ((m ₁ + m ₂ ) * v - m ₁ * v ₁ ) / m ₂ . After substitution of numerical values ​​and calculations, 25 m / s is obtained.

Answer. The speed of a small fragment is 25 m / s.

The task of shooting at an angle

Condition. A gun was mounted on a platform of mass M. A shot is fired from it with a shell of mass m. It flies out at an angle α to the horizon at a speed v (given relative to the ground). It is required to know the value of the platform speed after the shot.

Decision. In this problem, one can use the law of conservation of momentum in the projection onto the OX axis. But only if the projection of the external resultant forces is zero.

For the direction of the OX axis, you need to choose the side where the projectile will fly, and parallel to the horizontal line. In this case, the projections of gravity and the support reaction on OX will be zero.

The problem will be solved in a general way, since there is no specific data for the known quantities. The answer in it is the formula.

The momentum of the system before the shot was zero, because the platform and the projectile were stationary. Let the desired platform speed be denoted by the Latin letter u. Then its momentum after the shot is defined as the product of mass and projection of speed. Since the platform will roll back (against the direction of the OX axis), the momentum value will be with a minus sign.

Projectile momentum is the product of its mass and the projection of velocity on the OX axis. Due to the fact that the speed is directed at an angle to the horizon, its projection is equal to the speed times the cosine of the angle. In literal equality, it will look like this: 0 = - Mu + mv * cos α. From it, through simple transformations, we obtain the answer formula: u = (mv * cos α) / M.

Answer. The speed of the platform is determined by the formula u = (mv * cos α) / M.

The task of crossing the river

Condition. The width of the river along its entire length is the same and equal to l, its banks are parallel. The known velocity of the water flow in the river is v ₁ and the proper speed of the boat is v ₂ . 1). When crossing the bow of the boat is directed strictly to the opposite shore. How far does s carry it downstream? 2). At what angle α should the bow of the boat be directed so that it reaches the opposite shore strictly perpendicular to the point of departure? How much time t will be required for such a crossing?

Decision. 1). The full speed of the boat is the vector sum of two values. The first of them is the flow of the river, which is directed along the banks. The second is the boat’s own speed perpendicular to the shores. Two similar triangles are obtained in the drawing. The first is formed by the width of the river and the distance by which the boat blows. The second - velocity vectors.

The following record follows from them: s / l = v ₁ / v ₂ . After the conversion, the formula for the desired quantity is obtained: s = l * (v ₁ / v ₂ ).

2). In this version of the problem, the total velocity vector is perpendicular to the shores. It is equal to the vector sum of v ₁ and v ₂ . The sine of the angle by which the eigenvector should deviate is equal to the ratio of the modules v ₁ and v ₂ . To calculate the travel time, you will need to divide the width of the river by the calculated full speed. The value of the latter is calculated by the Pythagorean theorem.

v = √ (v ₂ ² - v ₁ ² ), then t = l / (√ (v ₂ ² - v ₁ ² )).

Answer. 1). s = l * (v ₁ / v ₂ ), 2). sin α = v ₁ / v ₂ , t = l / (√ (v ₂ ² - v ₁ ² )).