Physics and mathematics cannot do without the concept of "vector quantity". It must be known and recognized, as well as be able to operate with it. You should definitely learn this so as not to get confused and not make stupid mistakes.
How to tell a scalar value from a vector quantity?
The first one always has only one characteristic. This is its numerical value. Most scalars can take both positive and negative values. Examples are electrical charge, work, or temperature. But there are scalars that cannot be negative, such as length and mass.
A vector quantity, in addition to a numerical quantity, which is always taken modulo, is also characterized by a direction. Therefore, it can be depicted graphically, that is, in the form of an arrow, the length of which is equal to the modulus of the value directed in a certain direction.
When writing, each vector quantity is indicated by an arrow sign on the letter. If we are talking about a numerical value, then the arrow is not written or it is taken modulo.
What are the most commonly performed actions with vectors?
First, a comparison. They may or may not be equal. In the first case, their modules are the same. But this is not the only condition. They must also have the same or opposite directions. In the first case, they should be called equal vectors. In the second, they are opposite. If at least one of the specified conditions is not met, then the vectors are not equal.
Then comes addition. It can be done according to two rules: a triangle or a parallelogram. The first prescribes to postpone first one vector, then from its end 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 postponed from one point. Then build them to a parallelogram. The result of the action should be considered the diagonal of the parallelogram drawn from the same point.
If a vector quantity is subtracted from another, then they are again plotted from one point. Only the result will be a vector that matches the one from the end of the second to the end of the first.
What vectors are studied in physics?
There are as many as there are scalars. You can simply remember what vector quantities exist in physics. Or know the signs by which they can be calculated. For those who prefer the first option, such a table will come in handy. It contains the main vector physical quantities.
| Designation in the formula | Name |
| v | speed |
| r | move |
| a | acceleration |
| F | strength |
| r | impulse |
| E | electric field strength |
| B | magnetic induction |
| M | moment of force |
Now a little more about some of these quantities.
The first value is speed
It is worth starting to give examples of vector quantities from it. 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 specifies a numerical value and a direction. Therefore, speed is a vector quantity. In addition, it is customary to divide it into types. The first is linear speed. It is introduced when considering rectilinear uniform motion. At the same time, 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 to be chosen must necessarily be as short as possible. When the time interval tends to zero, the speed value is already instantaneous.
If arbitrary motion is considered, then here speed is always a vector quantity. After all, it has to be decomposed into components directed along each vector directing the coordinate lines. In addition, it is defined as the derivative of the radius vector, taken with respect to time.
The second value is strength
It determines the measure of the intensity of the impact that is exerted on the body by other bodies or fields. Since force is a vector quantity, it necessarily has its own modulo 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 vectors of forces, you can refer to the following table.
| Power | Application point | Direction |
| gravity | body center | to the center of the Earth |
| gravity | body center | to the center of another body |
| elasticity | point of contact between interacting bodies | against outside influence |
| friction | between touching surfaces | in the opposite direction of the movement |
Also, the resultant force is also a vector quantity. 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 you need to postpone the vectors in turn from the end of the previous one. The result will be the one that connects the beginning of the first to the end of the last.
Third value - movement
During the movement, the body describes a certain line. It's called a trajectory. This line can be completely different. More important is not its appearance, but the points of beginning and end of the movement. They connectsegment, which is called displacement. 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 customary to designate it with the Latin letter r.
Here the question may appear: "Is the path a vector quantity?". In general, this statement is not true. The path is equal to the length of the trajectory and has no definite direction. An exception is the situation when rectilinear movement in one direction is considered. Then the modulus of the displacement vector coincides in value with the path, and their direction turns out to be the same. Therefore, when considering movement along a straight line without changing the direction of movement, the path can be included in the examples of vector quantities.
The fourth value is acceleration
It is a characteristic of the rate of change of speed. Moreover, acceleration can have both positive and negative values. In rectilinear motion, it is directed in the direction of higher speed. If the movement occurs along a curvilinear trajectory, then its acceleration vector is decomposed into two components, one of which is directed towards the center of curvature along the radius.
Separate the average and instantaneous value of acceleration. The first should be calculated as the ratio of the change in speed over a certain period of time to this time. When the considered time interval tends to zero, one speaks of instantaneous acceleration.
The fifth magnitude is momentum
It's differentalso called momentum. 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 give it to the momentum.
By definition, the latter is equal to the product of body mass and speed. Using the concept of the momentum of a body, one can write the well-known Newton's law in a different way. It turns out that the change in momentum is equal to the product of force and time.
In physics, the law of conservation of momentum plays an important role, which states that in a closed system of bodies its total momentum is constant.
We have very briefly listed what quantities (vector) are studied in the course of physics.
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 wagon are 10 and 40 tons, respectively. The car hits the platform, an automatic coupler occurs. It is necessary to calculate the speed of the wagon-platform system after the impact.
Decision. First, you need to enter the notation: the speed of the car before impact - v1, the car with the platform after coupling - v, the weight of the car m1, the platform - m 2. According to the condition of the problem, it is necessary to find out the value of the speed v.
The rules for solving such tasks require a schematic representation of the system before and after the interaction. It is reasonable to direct the OX axis along the rails in the direction the car is moving.
Under these conditions, the system of wagons can be considered closed. This is determined by the fact that externalforces can be neglected. The force of gravity and the reaction of the support are balanced, and the 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 the coupler after the impact. At first, the platform did not move, so its momentum was zero. Only the car moved, its momentum is the product of m1 and v1.
Since the impact was inelastic, that is, the wagon grappled with the platform, and then it began to roll together in the same direction, the momentum of the system did not change direction. But its meaning has changed. Namely, the product of the sum of the mass of the wagon with the platform and the required speed.
You can write this equality: m1v1=(m1 + m2)v. It will be true for the projection of momentum vectors on the selected axis. From it it is easy to derive the equality that will be required to calculate the required speed: v=m1v1 / (m1 + m2).
According to the rules, you should convert values for mass from tons to kilograms. Therefore, when substituting them into the formula, you should 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.
Problem with dividing the body into parts
Condition. The speed of a flying grenade is 20 m/s. It breaks into two pieces. The mass of the first is 1.8 kg. It continues to move in the direction in which the grenade was flying at a speed of 50 m/s. The second fragment has a mass of 1.2 kg. What is its speed?
Decision. Let the fragment masses be denoted by the letters m1 and m2. Their speeds will respectively be v1 and v2. The initial speed of the grenade is v. In the problem, you need to calculate the value v2.
In order for the larger fragment to continue moving in the same direction as the whole grenade, the second must fly in the opposite direction. If we choose the direction of the axis as that of the initial impulse, then after the break, a large fragment flies along the axis, and a small fragment flies against the axis.
In this problem, it is allowed to use the law of conservation of momentum due to the fact that the explosion of a grenade occurs instantly. Therefore, despite the fact that gravity acts on the grenade and its parts, it does not have time to act and change the direction of the momentum vector with its modulo value.
The sum of the vector values of the momentum after the grenade burst is equal to the one before it. If we write the law of conservation of momentum of the body in projection onto the OX axis, then it will look like this: (m1 + m2)v=m 1v1 - m2v2. It is easy to express the desired speed from it. It is determined by the formula: v2=((m1 + m2)v - m 1v1) / m2. After substitution of numerical values and calculations, 25 m/s is obtained.
Answer. The speed of a small fragment is 25 m/s.
Problem about shooting at an angle
Condition. A tool is mounted on a platform of mass M. A projectile of mass m is fired from it. It flies out at an angle α tohorizon with a speed v (given relative to the ground). It is required to find out the value of the speed of the platform after the shot.
Decision. In this problem, you can use the momentum conservation law in projection onto the OX axis. But only in the case when the projection of the external resultant forces is equal to 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 the forces of gravity and the reaction of the support on OX will be equal to zero.
The problem will be solved in a general way, since there are no specific data for known quantities. The answer is the formula.
The momentum of the system before the shot was equal to zero, since the platform and the projectile were stationary. Let the desired speed of the platform be denoted by the Latin letter u. Then its momentum after the shot is determined as the product of the mass and the projection of the velocity. Since the platform rolls back (against the direction of the OX axis), the momentum value will be minus.
The momentum of a projectile is the product of its mass and the projection of its velocity onto the OX axis. Due to the fact that the velocity is directed at an angle to the horizon, its projection is equal to the velocity multiplied by the cosine of the angle. In literal equality, it will look like this: 0=- Mu + mvcos α. From it, by simple transformations, the answer formula is obtained: u=(mvcos α) / M.
Answer. Platform speed is determined by the formula u=(mvcos α) / M.
River Crossing Problem
Condition. The width of the river along its entire length is the same and equal to l, its banksare parallel. We know the speed of the water flow in the river v1 and the own speed of the boat v2. one). When crossing, the bow of the boat is directed strictly to the opposite shore. How far s will it be carried downstream? 2). At what angle α should the bow of the boat be directed so that it reaches the opposite bank strictly perpendicular to the point of departure? How much time t would it take to make such a crossing?
Decision. one). The full speed of the boat is the vector sum of the two quantities. The first of these is the course of the river, which is directed along the banks. The second is the own speed of the boat, perpendicular to the shores. The drawing shows two similar triangles. The first is formed by the width of the river and the distance that the boat carries. The second - with velocity vectors.
The following entry follows from them: s / l=v1 / v2. After the transformation, the formula for the desired value is obtained: s=l(v1 / v2).
2). In this version of the problem, the total velocity vector is perpendicular to the banks. It is equal to the vector sum of v1 and v2. The sine of the angle by which the own velocity vector must deviate is equal to the ratio of the modules v1 and v2. To calculate the travel time, you will need to divide the width of the river by the calculated total speed. The value of the latter is calculated using the Pythagorean theorem.
v=√(v22 – v1 2), then t=l / (√(v22 – v1 2)).
Answer. one). s=l(v1 / v2), 2). sin α=v1 /v2, t=l / (√(v22 – v 12)).
