The movement with the acceleration of various objects surrounds us every day. It is enough to recall cars, planes, bicycles, rotating wheels and shafts in order to understand the importance of accelerated movement of bodies in human life. There is a special section in physics to describe this motion, it is called kinematics. In this article, we consider what formulas describe the movement with uniformly accelerated motion.
Concept of acceleration, speed and path.
Before writing down the formulas for displacement with uniformly accelerated motion, we should give a concept to the main quantities that appear in them.
Let's start with the path. This value is understood as the distance that the body travels over an interval of time, moving along a known path. The shorter the body travels a certain path L, the greater its speed. Thus, the speed of the body is the speed of overcoming distances in space. At this time, the speed calculation is performed according to the following formula:
v¯ = dL / dt
Speed is a vector, and a path is a scalar. The speed is directed along the tangent restored to a given point of the trajectory.
If you observe a body moving along the trajectory for some time, and measure its speed at each point of the trajectory, it turns out that it is constantly changing. The change in speed is characterized by acceleration. In accordance with the definition of acceleration, it is calculated as follows:
a¯ = dv¯ / dt
Acceleration is also a vector quantity, only with the direction of speed it has nothing to do. The acceleration vector is turned in the direction of the change in speed for a given time or, which is the same thing, in the direction of the force acting on the body.
Equally Accelerated Motion in Physics
To understand what uniformly accelerated movement is, we give the following example: suppose that the car was at rest. Then he began to move, constantly increasing his speed. If for equal periods of time the increase in the vehicle speed module was the same, then we can talk about uniformly accelerated body movement. In other words, during the type of displacement under consideration, acceleration is a constant (a = const).
Do not think that movement with constant acceleration can only increase speed. As a result of this movement, the speed of the body can also decrease until it stops completely. This situation occurs when the vehicle is in the process of braking. In this case, the acceleration will be directed against the velocity vector.
In nature, the fall of bodies is a common movement with constant acceleration. Up to certain speeds, when the air resistance can be neglected, the free fall is uniformly accelerated, and the acceleration is 9.81 m / s 2 .
Change in speed when driving with constant acceleration
In contrast to acceleration, speed with uniformly accelerated motion is not a constant value. For simplicity, consider moving in a straight line. If the vectors of the quantities a¯ and v¯ coincide, the following formula for the velocity holds:
v = v 0 + a * t
This expression suggests that before the acceleration appeared, the body already had a certain velocity v 0 .
This formula shows that the speed increases linearly over time. The graph of the function v (t) is a straight line that intersects the y axis at a distance v 0 from the origin.
With uniformly accelerated motion, speed can also decrease linearly. For this, it is necessary that the acceleration and velocity vectors be opposite (car braking, free take-off of the body in height). For this case, you can write this expression:
v = v 0 - a * t
As in the previous case, the equality graph is a straight line, only the coefficient of its inclination to the x axis will not be positive, but negative.
Movement with uniformly accelerated movement
The path formula is uniquely obtained if we take the time integral of speed. In the case when the speed of the body increases, the following expression can be written for the path:
L = v 0 * t + a * t 2/2
It can be seen that the graph of the function L (t) is a parabola (its right branch). That is, the distance traveled increases rapidly over time.
If acceleration leads to a decrease in speed, then the formula for displacement with uniformly accelerated movement will take the form:
L = v 0 * t - a * t 2/2
The graph for this equation will also be a parabola, however its branch is gradually approaching a certain constant value. The latter corresponds to the distance traveled to the stop of a moving body.
Constant acceleration around the circle
In order to more fully characterize the topic, one should also give formulas for displacement with uniformly accelerated circular motion. Unlike rectilinear motion, this type of movement is described by angular quantities. Nevertheless, angular quantities are complete analogues of the corresponding linear characteristics.
For speed with uniformly accelerated motion of rotation, the following formula is valid:
ω = ω 0 + α * t
Here ω is the angular velocity, which is measured in radians per second (rad / s), α is the angular acceleration, it is measured in rad / s 2 .
An analogue of the path for rotation is the rotation angle θ. The formula is valid for him:
θ = ω 0 * t + α * t 2/2
Thus, with uniformly accelerated rotation, the kinematics formulas retain their form, but they already contain angular physical quantities.