An important topic studied in physics is the problem of moving bodies in space. They can move along straight and curved paths. One of the simplest types is movement with uniformly accelerated rectilinear motion. Let's consider it in the article.
The mechanical movement of bodies
The issue of changing spatial coordinates in the process of moving objects has been of interest to people since ancient times. So, the ancient Greek philosopher Archimedes believed that movement is not a natural form of existence of surrounding objects. Indeed, any abandoned item, in the end, stops. Archimedes did not know about the forces of friction, so he believed that the true state of all bodies is peace. This point of view was shared by another philosopher of Ancient Greece, Zeno, who in his famous paradoxes argued that movement is an illusion and does not really exist.
Only with the advent of the New Time did the theoretical study of the question of motion reach the next level. An important contribution to the study was made by the Italian Galileo Galilei. Newton’s first law was actually formulated by him. In his writings, he notes that every body will remain unchanged by the characteristics of its movement if no external forces act on it.
The theory of classical mechanical motion, expressed in concise and understandable mathematical formulas, was created by the great English scientist Isaac Newton. In particular, the second law bearing his name provides comprehensive information for describing the movement of bodies in space.
Strength, Acceleration and Mass
These three physical quantities are included in the famous formula known to every student. We write it down:
F¯ = m * a¯
Here, the force F¯ and acceleration a¯ are vector quantities directed in the same direction. The proportionality coefficient between them is the body mass m.
The recorded formula is the main expression of dynamics. Strength is the cause of changes in the motion characteristics of any body. The absence of an external force exerting an effect on the body is a sufficient condition for it to move uniformly and rectilinearly.
Further in the article, we consider the formulas of rectilinear uniformly accelerated motion, which describe only the result of the acting forces, that is, we will study the movement of bodies from the point of view of kinematics.
Constant speed
Before considering movement in a rectilinear uniformly accelerated motion, it is necessary to study its special case - uniform change of coordinates in space. Rectilinear uniform motion is mathematically described by the following equality:
S = v * t
That is, the path traveled by the body is proportional to the time during which it is in motion, and the speed of movement. Since the speed is measured in in the SI system in meters per second (m / s), and the time in seconds (s), the path is calculated in meters (m).
A feature of this type of movement is its straightforward nature, and the speed remains constant at each moment of time (v = const.). If we plot the graphs of speed and path versus time in a two-dimensional rectangular coordinate system, then in the first case we get a horizontal (parallel to the x axis) line, and in the second case, a straight line coming from the origin with an angular coefficient equal to speed (k = v)
Rectilinear, uniformly accelerated motion: acceleration and speed
Now suppose that a force began to act on a body at rest. According to Newton’s second law, it will cause the acceleration of the body. The speed in this case will vary according to the law:
v = a * t
This expression says that speed, in contrast to uniform motion, will no longer be a constant value. It grows linearly over time. In turn, the acceleration a is a constant, so the motion is called uniformly accelerated, that is, occurring with constant acceleration.
The graph of the speed of a rectilinear uniformly accelerated movement is a straight line that goes from the origin. Its angular coefficient k is equal to the acceleration a.
Equally accelerated motion with initial velocity
The formula for rectilinear uniformly accelerated motion written for the speed in the previous paragraph is not general. The fact is that when compiling it, it was assumed that the body is at rest at zero time, that is, v = 0. However, in the general case, it can move uniformly with a speed v 0 ≠ 0, which is called the initial one. In this case, the previous formula for speed can be rewritten as follows:
v = v 0 + a * t
The movement described by this expression can be represented as follows: a body, on which the forces did not act, moved with a speed v 0 . Then an external force appeared, directed in the same way as the initial velocity, and led to the appearance of acceleration a.
In the recorded expression, the time t is counted from zero. Therefore, when displaying the speed graph in this case, we get a similar straight line parallel to the straight line for uniformly accelerated movement without an initial speed. However, it will not start from the point (0; 0), but from the point (0; v 0 ).
Note that if the force acting on the body is directed against the initial velocity vector, then there is an equally slow motion. In this case, the following formula will be valid:
v = v 0 - a * t
The graph of this function is a line that tends to a zero value of v from the initial value of v 0 . The point where the graph crosses the x (t) axis will show the time when the body stops. This time is equal to:
t = v 0 / a
Distance traveled
Moving in a rectilinear uniformly accelerated motion means which path the body has traveled in a known period of time. This path can be calculated if we integrate the corresponding expressions for speed over time.
The path in a rectilinear uniformly accelerated motion, when the initial speed is absent, is determined from the expression:
S = ∫ t (v * dt) = a * t 2/2
Since the acceleration a is constant for the type of motion under consideration, the distance traveled increases as a square of time. The graph of uniformly accelerated rectilinear motion for path S is the right branch of the parabola, which begins at the origin.
Now we get a similar expression for the path S, we only take into account that before acceleration the body already had a speed v 0 . Then the formula for S takes the following form:
S = ∫ t (v * dt) = ∫ t ((v 0 + a * t) * dt) = v 0 * t + a * t 2/2.
The graph of this function S (t) will also be a parabola, however, its values will lie higher than in the previous case. A path with uniformly accelerated motion, when the initial speed is not equal to zero, can be considered as an overlap of two types of motion: uniform and uniformly accelerated.
Finally, if the body was moving at a speed of v 0 , and then it began to slow down its movement with constant acceleration a, then the formula for S will take the form:
S = ∫ t (v * dt) = ∫ t ((v 0 - a * t) * dt) = v 0 * t - a * t 2/2.
The figure below shows the graphs for the formulas written for S.
In the first case, we are talking about accelerated motion with a certain initial velocity, the third graph is accelerated motion without an initial velocity, and the second curve corresponds to equally slow motion. It can be seen that the two curves 1 and 3 increase constantly, but curve 2 reaches a constant value at some time t. This means that the body has stopped.
Acceleration Examples
In everyday life, we can cite a number of examples of movement in a rectilinear uniformly accelerated movement. For example, a car gaining speed after a start, an airplane that accelerates on the runway, a cyclist seeking to stop (equally slow).
Perhaps the most common movement in nature with constant acceleration is the fall of bodies in the gravitational field of our planet. If we neglect the force of air friction, then the process of falling near the surface of the Earth occurs with an acceleration of 9.81 m / s 2 .
Next, we show by examples of solving problems how to use the above formulas.
The task with a motorcyclist
It is known that the distance between cities is 50 km. A motorcyclist covered this distance in 1 hour. It is known that 1/3 of this time he moved with constant acceleration, and 2/3 of the time his speed did not change. It is necessary to calculate the final speed that the motorcyclist has developed.
To solve the problem, we use the following equations of rectilinear uniformly accelerated motion:
S = a * (1/3 * t) 2/2 + v * 2/3 * t;
v = a * 1/3 * t
Where t is the time the rider spent on the road. Substituting the second expression into the first, we obtain the acceleration value a:
S = a * (1/3 * t) 2/2 + a * 1/3 * t * 2/3 * t =>
a = 18 * S / (5 * t 2 ) = 18 * 50000 / (5 * 3600 * 3600) ≈ 0.014 m / s
Driving with this acceleration for 20 minutes, the motorcyclist developed speed:
v = a * 1/3 * t = 0.014 * 1/3 * 3600 = 16.8 m / s or about 60.5 km / h
Ball Thrown Up Challenge
The ball was thrown vertically up with an initial speed of 10 m / s. It is necessary to calculate how high it will rise.
We are talking about equally slow motion with an initial speed v 0 . If we denote the maximum lifting height h, then for it we can write the formula:
h = v 0 * t - g * t 2/2.
Here g = 9.81 m / s 2 , t is the rise time. To determine it, it is necessary to turn to the condition when the body velocity v becomes equal to zero:
v = v 0 - g * t = 0 =>
t = v 0 / g.
Substituting the time t in the formula for h, we obtain:
h = v 0 2 / g - g * v 0 2 / (2 * g 2 ) = v 0 2 / (2 * g).
We substitute here the speed value from the condition, we get the answer: h ≈ 5.1 meters. It is interesting to note that the height of the lifting of the body does not depend on its mass.