Movement is one of the important properties of the world around us. In physics, the motion of bodies is studied in a special section - kinematics. In this article, we consider what is called the acceleration of uniformly accelerated motion, and also by the example of solving the problem we will show how to calculate it.
The concept of acceleration
The movement of objects along various trajectories is described by such quantities as path, speed and acceleration. The concepts of path and speed are intuitively clear to every person. The mathematical formula connecting the path L and the velocity v¯ has the form
v¯ = dL / dt.
Speed is a vector quantity that is always directed along the tangent to the path of the bodies.
When answering the question of what is called the acceleration of uniformly accelerated motion, you must first deal with the physical concept itself. Acceleration is not as obvious as the two previous ones. According to the physical definition, it is understood as a change in speed over time, which mathematically can be written as follows:
a¯ = dv¯ / dt.
We note one important point in this definition: it refers to any change in speed that can manifest itself both in a change in its modulus and in direction.
Substituting the explicit expression for the velocity in the last formula, we can obtain another mathematical equality for acceleration through the second derivative of the path:
a¯ = d 2 L / dt 2 .
The acceleration in the SI system is measured in meters per square second (m / s 2 is abbreviated). So, the value of 1 m / s 2 means that the speed of the body increases by 1 m / s for every second.
Uniform acceleration and acceleration
What is called acceleration of uniformly accelerated motion? You can understand this if you consider the specified type of movement.
According to the name, uniformly accelerated movement is a movement of a body in space at which its acceleration does not change, that is, during the entire process of movement, it remains constant both in magnitude and in direction. Thus, the acceleration of a body during its uniformly accelerated movement is the quantity a, which can be obtained from the following expression:
a = Δv / Δt.
Here Δv is the absolute change in speed over a period of time Δt. The difference between this formula and the one written in the previous paragraph is that it calculates the average acceleration, not instantaneous. In addition, with uniformly accelerated motion, it makes no sense to use vector quantities, since the acceleration itself remains unchanged in direction (for rectilinear motion).
If the motion is curved, then the direction of acceleration changes, but the formula above remains valid anyway, since it describes the so-called tangential component of acceleration.
Rectilinear acceleration
This movement occurs in a straight line. In the general case, the path traveled by the body during time t is calculated by the formula
L = v 0 × t + a × t 2/2.
The value of a is called the acceleration of the body in a rectilinear uniformly accelerated motion. The value of v 0 is the speed that the body had before the time t began.
The given kinematic equation of motion allows one to calculate the acceleration if the time instant t and the path L that the body traveled to this moment are known. The sought expression has the form
a = 2 (L - v 0 × t) / t 2 .
An example of uniformly accelerated movement is the acceleration of a car or cyclist after starting. The acceleration vector in the case considered coincides with the velocity vector.
Circular motion with acceleration
In addition to the rectilinear movement in technology and nature, moving objects around a circle is often found. It can be both uniform, for example, the rotation of planets in its orbits, and accelerated, for example, the rotation of shafts and gears of mechanical machines.
What is called the acceleration of uniformly accelerated body movement in a circle? It is customary to calculate according to the following formula:
α = Δω / Δt.
The angular acceleration α at uniformly accelerated rotation is a constant value equal to the ratio of the change in the angular velocity Δω to the time interval Δt during which this change is observed. Unlike the linear acceleration discussed above, the value of α is measured in radians per square second (rad / s 2 ).
The kinematic equation of motion for uniformly accelerated rotation has the form
θ = ω 0 × t + α × t 2/2,
where θ is the angle in radians, ω 0 is the initial angular velocity.
The formula that relates linear (tangential) acceleration to angular, has the form
a = α × r.
This expression explains why it is convenient to use the angular characteristic α rather than the linear value a when rotating the bodies. While α is constant, a depends on the distance r from the axis of rotation.
The solution to the problem of determining acceleration
Having considered the question of what is called acceleration of uniformly accelerated movement, we will solve the following problem: a car, starting from a place on a straight road, traveled a path 100 meters more in the first 10 seconds than in the first 5 seconds. What acceleration did he move with?
To begin, we write the working formula:
L = a × t 2/2.
This expression follows from the above formula in the article for the path with uniformly accelerated rectilinear motion, given that the initial speed v 0 is zero.
Suppose that during time t 1 the car passed the path L 1 , and for time t 2 - the path L 2 . Then you can write:
L 1 = a × t 1 2/2.
L 2 = a × t 2 2/2.
Subtracting the first equality from the second and substituting the values from the conditions of the problem, we obtain:
L 2 - L 1 = a / 2 (t 2 2 - t 1 2 ) =>
a = 2 (L 2 - L 1 ) / (t 2 2 - t 1 2 ) ≈ 2.67 m / s 2 .
Thus, for every second of its movement, the car increased its speed by 2.67 m / s.