Types of motion (uniform, uniformly accelerated) and their graphic description
According to the shape of the trajectory, the movement is divided into curvilinear(the trajectory of the body is a curved line) and rectilinear(trajectory of body movement is a straight line).
When a body moves along a straight path, the magnitude of the displacement vector always coincides with the path traveled. When a body moves along a curved path, the magnitude of the displacement vector is always less than the distance traveled
Uniform straight motion.
Rectilinear uniform movement is a movement in which a body makes equal movements in any equal intervals of time.
Speed of uniform rectilinear motion - this is a physical vector quantity equal to the ratio of the movement of the body S over any period of time to the value of this interval t:
v x =S/t
Speed - this is a physical quantity that shows the speed of change of coordinates.
Speed units are meters per second
Equation of uniform motion (body movement with uniform motion) :
S= v x t
Body coordinate equation:
x=x 0 + v x t
Designations:
X- coordinate of the moving body
x 0- initial coordinate of the moving body
vWed -Average speed of uniform linear motion
v X- Speed of uniform linear motion
S - Body displacement (distance the body moved)
t - Time interval of movement (time)
Graphical representation of uniform linear motion
v
Dependence of acceleration on time. Since during uniform motion the acceleration is zero, the dependence a(t) is a straight line that lies on the time axis.
Since the body moves rectilinearly and uniformly ( v =const), i.e. the speed does not change over time, then the graph with the dependence of speed on time v (t) is a straight line parallel to the time axis.
The projection of the body's displacement is numerically equal to the area of the rectangle under the graph, since the magnitude of the displacement vector is equal to the product of the velocity vector and the time during which the displacement was made.
in case of rectilinear uniform motion, the magnitude of the displacement vector is equal to the area of the rectangle under the velocity graph.
Dependence of displacement on time. Graph s(t) - sloping line :
Dependence of coordinates on time. Graph x(t) - sloping line :
The graph shows that the projection of the velocity is equal to:
v x =S/t=tga
Having considered this formula, we can say that the larger the angle a, the faster the body moves and it covers a greater distance in less time.
The rule for determining speed from the graph of s(t) and x(t): The tangent of the angle of inclination of the graph to the time axis is equal to the speed of movement.
Uneven straight motion.
Uniform motion is motion at a constant speed. If the speed of a body changes, it is said to be moving unevenly.
A movement in which a body makes unequal movements at equal intervals of time is called uneven or variable movement.
To characterize uneven motion, the concept of average speed is introduced.
Average driving speed equal to the ratio of the entire path traveled by a material point to the period of time during which this path was traveled.
In physics, the greatest interest is not the average, but instantaneous speed , which is defined as the limit to which the average speed tends over an infinitesimal period of time Δ t:
Instant speedvariable motion is the speed of a body at a given point in time or at a given point on the trajectory.
The instantaneous velocity of a body at any point on a curvilinear trajectory is directed tangentially to the trajectory at that point.
The difference between average and instantaneous speeds is shown in the figure.
The movement of a body in which its speed changes equally over any equal periods of time is calleduniformly accelerated or uniformly alternating motion.
Acceleration -this is a vector physical quantity that characterizes the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.
If the speed changes equally throughout the entire movement, then the acceleration can be calculated using the formula:
Designations:
v x - final speed of a body during uniformly accelerated motion in a straight line
v 0x - initial speed of the body
a- body acceleration
t - time of body movement
Acceleration shows how quickly the speed of a body changes. If the acceleration is positive, then the speed of the body increases, the movement is accelerated. If the acceleration is negative, it means the speed is decreasing and the movement is slow.
SI unit of acceleration [ m/s 2 ].
Acceleration is measured accelerometer
Speed equation for uniformly accelerated motion:
Equation of uniformly accelerated rectilinear motion(movement during uniformly accelerated motion):
Designations:
Displacement of a body during uniformly accelerated motion in a straight line
Initial speed of the body
Speed of a body during uniformly accelerated motion in a straight line
Body acceleration
Body movement time
More formulas for finding displacement during uniformly accelerated linear motion, which can be used when solving problems:
- if the initial and final speeds and acceleration are known.
- if the initial, final speeds of movement and the time of the entire movement are known
Graphical representation of uneven linear motion
Mechanical movement is represented graphically. The dependence of physical quantities is expressed using functions. Designate:
v (t) - change in speed over time
S(t) - change in displacement (path) over time
a(t) - change in acceleration over time
Dependence of acceleration on time. Acceleration does not change with time, has a constant value, the graph a(t) is a straight line parallel to the time axis.
Dependence of speed on time. With uniform motion, the speed changes according to a linear relationship .
The graph is a sloping line.
The rule for determining the path using the graph v(t): The path of a body is the area of the triangle (or trapezoid) under the velocity graph.
The rule for determining acceleration using the graph v(t): The acceleration of a body is the tangent of the angle of inclination of the graph to the time axis. If the body slows down, the acceleration is negative, the angle of the graph is obtuse, so we find the tangent of the adjacent angle.
Dependence of the path on time. With uniformly accelerated motion, the path changes according to the quadratic relationship
In coordinates, the dependence has the form 
.
The graph is a branch of a parabola.
Mechanical movement is considered for material point and For solid body.
Motion of a material point
Forward movement an absolutely rigid body is a mechanical movement during which any straight line segment associated with this body is always parallel to itself at any moment in time.
If you mentally connect any two points of a rigid body with a straight line, then the resulting segment will always be parallel to itself in the process of translational motion.
During translational motion, all points of the body move equally. That is, they travel the same distance in the same amount of time and move in the same direction.
Examples of translational motion: the movement of an elevator car, mechanical scales, a sled rushing down a mountain, bicycle pedals, a train platform, engine pistons relative to the cylinders.
Rotational movement
During rotational motion, all points of the physical body move in circles. All these circles lie in planes parallel to each other. And the centers of rotation of all points are located on one fixed straight line, which is called axis of rotation. Circles that are described by points lie in parallel planes. And these planes are perpendicular to the axis of rotation.
Rotational movement is very common. Thus, the movement of points on the rim of a wheel is an example of rotational movement. Rotational motion is described by a fan propeller, etc.
Rotational motion is characterized by the following physical quantities: angular velocity of rotation, period of rotation, frequency of rotation, linear speed of a point.
Angular velocity A body rotating uniformly is called a value equal to the ratio of the angle of rotation to the period of time during which this rotation occurred.
The time it takes a body to complete one full revolution is called rotation period (T).
The number of revolutions a body makes per unit time is called speed (f).
Rotation frequency and period are related to each other by the relation T = 1/f.
If a point is located at a distance R from the center of rotation, then its linear speed is determined by the formula:
The point is that, when considering a particular body, one should take into account that all its points move in the same direction with absolutely the same speed. That is why it is not necessary to characterize the movement of the entire given body; you can limit yourself to just one point.
The main characteristics of any movement include its trajectory, movement and speed. A trajectory is just a line that exists only in the imagination, along which a given material point moves in space. The displacement is a vector directed from the starting point to the ending point. Finally, speed is a general indicator of the movement of a point, which characterizes not only its direction, but also the speed of movement relative to any body taken as a reference point.
Uniform linear motion is a largely imaginary concept that is characterized by two main factors - uniformity and straightness.
Uniformity of movement means that it is carried out at a constant speed without any acceleration. Straightness of movement implies that it occurs along a straight line, that is, its trajectory is an absolutely straight line.
Based on all of the above, we can conclude that uniform linear motion is a special type of motion, as a result of which the body performs the same movement in absolutely equal periods of time. Thus, by dividing a certain interval into equal intervals (for example, one second), it will be possible to see that with the movement indicated above, the body will cover the same distance for each of these segments.
The speed of uniform rectilinear motion is which, in numerical terms, is equal to the ratio of the path traveled by the body in a given period of time to the numerical value of this interval. This value does not depend in any way on time; moreover, it is worth noting that the speed of uniform rectilinear motion at any point of the trajectory absolutely coincides with the movement of the body. In this case, the quantitative value for an arbitrarily taken period of time is equal to
Uniform linear motion is characterized by a special approach to the path that the body travels over a certain period of time. The distance traveled in this case is nothing more than a displacement module. Displacement, in turn, is the product of the speed with which the body moved and the time during which this movement was carried out.
It is quite natural that if the displacement vector coincides with the positive direction of the x-axis, then the projection of the calculated velocity will not only be positive, but also coincide with the magnitude of the velocity.
Uniform rectilinear motion can be represented, among other things, in the form of an equation, which will reflect the relationship between the coordinates of the body and time.
To find the coordinates of a moving body at any moment in time, you need to know the projections of the displacement vector on the coordinate axes, and therefore the displacement vector itself. What you need to know for this. The answer depends on what kind of movement the body makes.
Let's first consider the simplest type of movement - rectilinear uniform motion.
A movement in which a body makes equal movements at any equal intervals is called rectilinear uniform movement.
To find the displacement of a body in uniform rectilinear motion over a certain period of time t, you need to know what movement a body makes per unit of time, since for any other unit of time it makes the same movement.
The movement made per unit of time is called speed body movements and are designated by the letter υ . If movement in this area is denoted by , and the time period is denoted by t, then the speed can be expressed as a ratio to . Since displacement is a vector quantity, and time is a scalar quantity, then speed is also a vector quantity. The velocity vector is directed in the same way as the displacement vector.
Speed of uniform linear motion of a body is a quantity equal to the ratio of the movement of the body to the period of time during which this movement occurred:
Thus, speed shows how much movement a body makes per unit time. Therefore, to find the displacement of a body, you need to know its speed. The movement of the body is calculated by the formula:
The displacement vector is directed in the same way as the velocity vector, time t- scalar quantity.
Calculations cannot be carried out using formulas written in vector form, since a vector quantity has not only a numerical value, but also a direction. When making calculations, they use formulas that include not vectors, but their projections on the coordinate axes, since algebraic operations can be performed on projections.
Since the vectors are equal, their projections onto the axis are also equal X, from here:
Now you can get a formula for calculating the coordinates x points at any given time. We know that
From this formula it is clear that with rectilinear uniform motion, the coordinate of the body linearly depends on time, which means that with its help it is possible to describe rectilinear uniform motion.
In addition, it follows from the formula that to find the position of the body at any time during rectilinear uniform motion, you need to know the initial coordinate of the body x 0 and the projection of the velocity vector onto the axis along which the body moves.
It must be remembered that in this formula v x- projection of the velocity vector, therefore, like any projection of a vector, it can be positive and negative.
Rectilinear uniform motion is rare. More often you have to deal with movement in which the movements of the body can be different over equal periods of time. This means that the speed of the body changes somehow over time. Cars, trains, airplanes, etc., a body thrown upward, and bodies falling to the Earth move at variable speeds.
With such a movement, you cannot use a formula to calculate the displacement, since the speed changes over time and we are no longer talking about a specific speed, the value of which can be substituted into the formula. In such cases, the so-called average speed is used, which is expressed by the formula:
average speed shows the displacement that a body makes on average per unit of time.
However, using the concept of average speed, the main problem of mechanics - determining the position of a body at any moment in time - cannot be solved.
1) Analytical method.We consider the highway to be straight. Let's write down the equation of motion of a cyclist. Since the cyclist moved uniformly, his equation of motion is:
(we place the origin of coordinates at the starting point, so the initial coordinate of the cyclist is zero).
The motorcyclist was moving at uniform acceleration. He also started moving from the starting point, so his initial coordinate is zero, the initial speed of the motorcyclist is also zero (the motorcyclist began to move from a state of rest).
Considering that the motorcyclist started moving later, the equation of motion for the motorcyclist is:
In this case, the speed of the motorcyclist changed according to the law:
At the moment when the motorcyclist caught up with the cyclist, their coordinates are equal, i.e. or:
Solving this equation for , we find the meeting time:
This is a quadratic equation. We define the discriminant:
Determining the roots:
Let's substitute numerical values into the formulas and calculate:
We discard the second root as not corresponding to the physical conditions of the problem: the motorcyclist could not catch up with the cyclist 0.37 s after the cyclist started moving, since he himself left the starting point only 2 s after the cyclist started.
Thus, the time when the motorcyclist caught up with the cyclist:
Let's substitute this time value into the formula for the law of change in speed of a motorcyclist and find the value of his speed at this moment:
2) Graphic method.
On the same coordinate plane we build graphs of changes over time in the coordinates of the cyclist and motorcyclist (the graph for the cyclist’s coordinates is in red, for the motorcyclist – in green). It can be seen that the dependence of the coordinate on time for a cyclist is a linear function, and the graph of this function is a straight line (the case of uniform rectilinear motion). The motorcyclist was moving with uniform acceleration, so the dependence of the motorcyclist’s coordinates on time is a quadratic function, the graph of which is a parabola.