Mechanical movement has been around us since birth. Every day we see how cars move on roads, ships move on seas and rivers, planes fly, even our planet moves, crossing outer space. An important characteristic for all types of motion without exception is acceleration. This is a physical quantity, types and main characteristics of which will be considered in this article.
Physical Acceleration Concept
To many, the term “acceleration” is intuitively familiar. In physics, acceleration is a value that characterizes any change in speed over time. The corresponding mathematical formulation is:
a¯ = dv¯ / dt
The bar over the symbol in the formula means that this value is vectorial. Thus, the acceleration a¯ is a vector and it also describes the change in the vector quantity - the velocity v¯. This acceleration is called full, it is measured in meters per second square. For example, if the body increases speed by 1 m / s for every second of its movement, then the corresponding acceleration is 1 m / s 2 .
Where does acceleration come from and where is it directed?
We figured out the definition of acceleration. It was also found that we are talking about the magnitude of the vector. Where is this vector directed?
To give the correct answer to the question posed above, we should recall Newton’s second law. In conventional form, it is written as follows:
F¯ = m * a¯
Words can read this equality as follows: the force F¯ of any nature acting on a body of mass m leads to the appearance of acceleration a¯ of this body. Since mass is a scalar quantity, it turns out that the force and acceleration vectors will be directed along the same straight line. In other words, acceleration is always directed towards the action of the force and is completely independent of the velocity vector v¯. The latter is directed along the tangent to the motion path.
Curvilinear motion and full acceleration components
In nature, we often encounter the movement of bodies along curved paths. Let us consider how acceleration can be described in this case. To do this, suppose that the velocity of a material point in the considered part of the trajectory can be written in the form:
v¯ = v * u t ¯
The velocity v¯ is the product of its absolute value v and the unit vector u t ¯ directed along the tangent to the trajectory (tangential component).
By definition, acceleration is a derivative of speed with respect to time. We have:
a¯ = dv¯ / dt = d (v * u t ¯) / dt = dv / dt * u t ¯ + v * d (u t ¯) / dt
The first term on the right side of the written equality is called tangential acceleration. Like speed, it is directed along the tangent and characterizes a change in the absolute value of v¯. The second term is the normal acceleration (centripetal), it is directed perpendicular to the tangent and characterizes the change in the vector of the quantity v¯.
Thus, if the radius of curvature of the trajectory is equal to infinity (straight line), then the velocity vector in the process of moving the body does not change its direction. The latter means that the normal component of the full acceleration is zero.
In the case of a material point moving around the circle uniformly, the velocity modulus remains constant, that is, the tangential component of full acceleration is zero. The normal component is directed to the center of the circle and is calculated by the formula:
a n = v 2 / r
Here r is the radius. The reason for the appearance of centripetal acceleration is the action on the body of a certain internal force, which is directed toward the center of the circle. For example, for the movement of planets around the sun, this force is gravitational attraction.
The formula that relates the absolute acceleration modules and its components a t (tangent), a n (normal) has the form:
a = √ (a t 2 + a n 2 )
Equal acceleration in a straight line
The movement in a straight line with constant acceleration is often found in everyday life, for example, this is the movement of a car on the road. This type of motion is described by the following equation for speed:
v = v 0 + a * t
Here v 0 is a certain speed that the body possessed before the acceleration a appeared in it.
If we plot the graph of the function v (t), then we get a straight line that the y axis intersects at the point with coordinates (0; v 0 ), and the tangent of the angle of inclination to the x axis is equal to the acceleration modulus a.
Taking the integral of the function v (t), we obtain the formula for the path L:
L = v 0 * t + a * t 2/2
The graph of the function L (t) is the right branch of the parabola, which begins at the point (0; 0).
The above formulas are the basic equations of kinematics of accelerated movement in a straight line.
If the body, having an initial speed v 0 , begins to slow down its movement with constant acceleration, then they speak of equally slow movement. The following formulas are valid for him:
v = v 0 - a * t;
L = v 0 * t - a * t 2/2
The solution to the problem of calculating acceleration
While stationary, the car begins to move. At the same time, in the first 20 seconds, it passes a distance of 200 meters. What is the acceleration of a car?
First, we write the general kinematic equation for the path L:
L = v 0 * t + a * t 2/2
Since in our case the vehicle was at rest, its speed v 0 was equal to zero. We get the formula for acceleration:
L = a * t 2/2 =>
a = 2 * L / t 2
We substitute the value of the distance traveled L = 200 m for a period of time t = 20 s and write down the answer to the question of the problem: a = 1 m / s 2 .