The basis of many problems in physics is the consideration of rectilinear uniform and uniformly accelerated motion. They are the simplest and idealized cases of the movement of bodies in space. We describe them in more detail in this article.
What is movement?
Before considering uniform and uniformly accelerated rectilinear motion, it is useful to deal with the concept itself.
Movement is a process of changing the coordinates of a material point in space for a certain period of time. According to this definition, we distinguish the following features by which you can immediately say whether it is a question of movement or not:
- There must be a change in spatial coordinates. Otherwise, the body can be considered as resting.
- The process should evolve over time.
Also pay attention to the concept of "material point". The fact is that when studying the issues of mechanical motion (including uniform and equally accelerated rectilinear motion, including), the body structure and its dimensions are not taken into account. This approximation is connected with the fact that the magnitude of the change in the coordinates in space far exceeds the physical dimensions of a moving object, therefore it is considered a material point (the word "material" involves taking its mass into account, since its knowledge is necessary when solving the problems under consideration).
The main physical quantities characterizing the movement
These include speed, acceleration, the distance traveled, as well as the concept of a trajectory. We analyze each value in order.
The speed of rectilinear uniform and uniformly accelerated motion (vector value) reflects the speed of change of the coordinates of the body in time. For example, if it moved in 10 seconds to 100 meters (typical values for sprinters in sports), then they talk about a speed of 10 meters per second (100/10 = 10 m / s). This value is denoted by the Latin letter "v" and is measured in units of distance divided by time, for example, kilometers per hour (km / h), meters per minute (m / min.), Miles per hour (mph) and so on. Further.
Acceleration is a physical vector quantity, which is indicated by the letter "a", and is characterized by the speed of change of speed itself. Returning to the example of sprinters, it is known that at the start of a race they start at a slow speed, as it moves it increases, reaching maximum values. The dimension of acceleration is obtained if we divide it for speed by time, for example, (m / s) / s or m / s 2 .
The distance traveled (scalar value) reflects the distance that a moving object traveled (traveled, flew, swam). This value is uniquely determined only by the initial and final position of the object. It is measured in units of distance (meters, kilometers, millimeters and others) and is indicated by the letter "s" (sometimes "d" or "l").
The trajectory, in contrast to the path, characterizes the curved line along which the body moved. Since in this article we consider only motion uniformly accelerated and uniformly rectilinear, then the trajectory for it will be a straight line.
The issue of relativity
Many people noticed that while on the bus, you can see that the car moving in the next lane seems to be at rest. This example clearly confirms the relativity of movement (uniformly accelerated, uniform rectilinear movement and its other types).
Given this feature, when considering tasks with moving objects, a reference system is always introduced, with respect to which the problem is solved. So, if we take the passenger in the bus for the reporting system in the example above, then the speed of the car relative to it will be zero. If we consider the movement relative to the person standing at the stop, then the car moves with respect to him at a certain speed v.
In the case of rectilinear movement, when two objects move along one line, the speed of one of them relative to the other is determined by the formula: v ¯ = v ¯ 1 + v ¯ 2 , here v ¯ 1 and v ¯ 2 are the speeds of each object (bar means that the vector quantities add up).
The easiest way to move
Of course, such is the movement of an object in a straight line with a constant speed (uniform rectilinear). An example of this type of movement is an airplane flying through the clouds or walking a pedestrian. In both cases, the trajectory of the object remains straight, and each of them moves with a specific speed.
Formulas describing this type of movement of objects have the following form:
Here t is the period of time during which the movement is considered.
Equally accelerated rectilinear movement
It is understood as a type of rectilinear movement of an object in which its speed changes according to the formula v = a * t, where a is constant acceleration. The change in speed arises due to the action of external forces of different nature. For example, the same plane, before it reaches cruising speed, must pick it up from a state of rest. Another example: car braking when the speed changes from a certain value to zero. This type of motion is called equally slow because acceleration has a negative sign in it (directed against the velocity vector).
The distance s traveled for this type of displacement can be calculated if we integrate the velocity over time, as a result we get the formula: s = a * t 2/2, where t is the acceleration (braking) time.
Mixed Movement
In some cases, the rectilinear movement of objects in space occurs both with constant speed and with acceleration, therefore it is useful to give formulas for this mixed type of motion.
The speed and acceleration of uniform and uniformly accelerated rectilinear motion are related to each other by the following expression: v = v 0 + a * t, where v 0 is the value of the initial speed. To understand this formula is simple: at first the object moved with a constant speed v 0 , for example, a car on the road, but then it began to accelerate, that is, for each period of time t, it began to increase the speed of its movement by a * t. Since the velocity is an additive quantity, the sum of its initial value with the magnitude of the change will lead to the noted expression.
Integrating this formula over time, we obtain another equation of rectilinear uniform and uniformly accelerated motion, which allows us to calculate the distance traveled: s = v 0 * t + a * t 2/2. As you can see, this expression is equal to the sum of similar formulas for simpler types of motion, discussed in the previous paragraphs.
Problem solving example
We solve a simple problem that will demonstrate the use of the above formulas. The condition for the task is as follows: the car, moving at a speed of 60 km / h, began to brake and after 10 seconds completely stopped. What path did he go during braking?
In this case, we are dealing with a rectilinear, equally slow motion. The initial speed v 0 = 60 km / h, the final value of this value is v = 0 (the car stopped). To determine the acceleration of braking, we use the formula: v = v 0 - a * t (the "-" sign says that the body slows down). We translate km / h to m / s (60 km / h = 16.667 m / s), and taking into account that the braking time is t = 10 s, we get: a = (v 0 - v) / t = 16.667 / 10 = 1.667 m / s 2 . We determined the braking acceleration of the car.
To calculate the distance traveled, we also use the equation for the mixed type of motion taking into account the sign of acceleration: s = v 0 * t - a * t 2/2. Substituting the known values, we obtain: s = 16.667 * 10 - 1.667 * 10 2/2 = 83.33 meters.
Note that the distance traveled could be found using the formula for uniformly accelerated movement (s = a * t 2/2), because when braking, the car will travel exactly the same distance as during acceleration from a standstill to reaching speed v 0 .
Curve movement
It is important to note that the considered expressions for the path traveled are applicable not only for the case of rectilinear motion, but also for any movement of the object along a curved path.
For example, to calculate the distance that our planet will fly around the Sun (moving in a circle) over a certain period of time, we can successfully use the expression s = v * t. This can be done because it uses the speed module, which is a constant value, but the velocity vector changes. Applying the formula for the path along a curved path, it should be borne in mind that the obtained value will reflect the length of this path, and not the difference between the final and initial coordinates of the object.