The issue of moving bodies in space is an object of study of kinematics. From the point of view of this section of physics, we consider in this article a special type of displacement — uniformly accelerated motion. Definition and formulas will be given below.
Trajectory, path, speed and acceleration
Before considering the definition of uniformly accelerated motion, one should introduce the main characteristics and concepts that are used in kinematics to describe the mechanical movement of bodies. The title of the paragraph contains these concepts. Let us describe each of them in more detail.
The trajectory of movement is an imaginary, non-existent line of space along which the body carries out movement. The simplest cases are rectilinear and circular paths, which are often found in nature and technology. For example, a car moving along a road performs straightforward motion. However, the same car, making a protracted turn, moves along a circular path of a certain radius.
The path (denoted by the letter L) is the distance, measured in units of length, that the body travels over the considered period of time. In the case of rotation, instead of the linear path, its angular analogue is used - the central angle of rotation in radians (we denote it by θ).
Speed is a vector motion characteristic that reflects the speed of change of spatial coordinates. Instantaneous speed, that is, speed at a given time, is calculated by the formula:
This value is measured in meters per second in SI and is always directed tangentially to the trajectory.
Acceleration is one of the important kinematics. It determines the speed of change of speed. Here, by change is understood not only the absolute value of v¯, but also its direction. According to this definition, one can write the following expression for acceleration:
The acceleration vector does not coincide with the velocity vector in the general case. It is directed towards the force acting on the body, therefore it has nothing to do with speed.
The definition of uniformly accelerated motion below is related to the behavior of the function a¯ (t).
Full acceleration and components of its decomposition
The shape of the path of bodies is a key issue in understanding the definition of uniformly accelerated motion. The fact is that any curvilinear trajectory assumes that the acceleration quantity a¯ has two nonzero components. They are called tangent and normal components of acceleration. The first determines the change in speed modulo (a t ), the second determines the change in speed in the direction (a c ). Both components are mutually perpendicular to each other, therefore, through their modules, the full acceleration can be calculated as follows:
It is important to remember that normal acceleration a c (also called centripetal) appears only in the case of a curved path. This acceleration does not depend on a change in the speed modulus.
Constant Acceleration
The title of this section is a definition of uniformly accelerated movement. In other words, the movement will be uniformly accelerated only when the acceleration is a constant and does not depend on time. What kind of acceleration are we talking about? Of course, about the tangent component of acceleration.
To explain why time independence of full acceleration does not guarantee that the motion is uniformly accelerated, we give a simple example. Suppose that a body rotates uniformly around a circle, for example, our Earth - around the Sun. The velocity modulus is stored at any point on the trajectory, which means that the tangential acceleration is zero, that is, the motion is uniform. Full acceleration is not equal to zero, since its normal component has some constant final value.
In the case of movement in a straight line, the tangent and full accelerations are equal to each other. This fact allows us to talk about full acceleration in the study of uniformly accelerated motion in a straight line.
Pure, uniformly accelerated motion in a straight line
By the word "pure" is meant that when considering this movement there is no initial velocity. The body from a state of rest immediately begins to move uniformly accelerated. In this case, full acceleration is constant. For speed, you can write the formula:
The graph of this equality is a line passing through a point (v = 0; t = 0). The tangent of the angle of inclination of the line to the x axis is numerically equal to the acceleration a.
Integrating the recorded expression over time, we obtain the formula for moving the body with uniformly accelerated motion:
The graph of the path L from time to time is the right branch of the parabola, which begins at the point (0; 0).
Constant Acceleration and Initial Speed
The question of uniformly accelerated motion without initial velocity was considered above. Now suppose that the body began to move and for some time moved with speed v 0 . Then, as a result of the action of an external force, it began to accelerate with constant acceleration a. In this case, the formula for speed will take the form:
As in the previous case, the graph of the function v (t) is also a straight line, only now it starts at the point (v 0 ; 0). Note that a report of time t begins to be made at the time of acceleration. So, if we substitute the value t = 0 in the formula, then we will get the initial velocity.
Taking the time integral of the function v (t), we obtain the corresponding expression for the path:
The graph of the function L (t) is a parabola, as in the case of pure motion with acceleration. This parabola also passes through the origin. Its only and important difference is that the branch of this parabola lies closer to the y axis than the previous graph (provided that the values of the accelerations a are equal in both cases).
Free fall in a gravitational field
We gave a definition of uniformly accelerated motion and considered the kinematics formulas with which it is described. Now we give the most famous example in nature of this type of movement - free fall.
On any body that has mass and is located near the surface of the earth, the force of gravitational attraction of our planet acts. This force is directly proportional to the mass, where the role of the proportionality coefficient is played by the acceleration of gravity. It is customary to denote the letter g. It is equal to approximately 9.81 m / s 2 , that is, for every second of a fall, the body increases its speed by almost 10 m / s.
Acceleration g is the "calling card" of each planet and depends on the radius and mass of the latter. So, on the Moon g = 1.62 m / s 2 , and on Jupiter it is 24.79 m / s 2 .
Challenge with a moving car
Starting a uniformly accelerated movement, the car moved for a while. During this time, he traveled a distance of 250 meters. Assuming that the acceleration is constant and equal to 2 m / s 2 , it is necessary to determine the time of movement of the vehicle.
This problem is solved using the formula for pure uniformly accelerated motion. We write it down:
We express the time t from here, we get:
The data from the condition lead to the answer: t = 15.8 seconds.