Mechanical movement surrounds us from birth. Every day we see how cars are moving along the roads, ships are moving along the seas and rivers, airplanes are flying, even our planet is moving, crossing outer space. An important characteristic for all types of movement without exception is acceleration. This is a physical quantity, the types and main characteristics of which will be discussed in this article.
Physical concept of acceleration
Many of the term "acceleration" is intuitively familiar. In physics, acceleration is a quantity that characterizes any change in speed over time. The corresponding mathematical formulation is:
a¯=dv¯/ dt
The line above the symbol in the formula means that this value is a vector. Thus, the acceleration a¯ is a vector and it also describes the change in a vector quantity - the speed v¯. This isacceleration is called full, it is measured in meters per square second. For example, if a body increases speed by 1 m/s for every second of its movement, then the corresponding acceleration is 1 m/s2.
Where does acceleration come from and where does it go?
We figured out the definition of what is acceleration. It was also found out that we are talking about the magnitude of the vector. Where is this vector pointing?
To give the correct answer to the above question, one should remember Newton's second law. In the common form, it is written as follows:
F¯=ma¯
In words, this equality can be read as follows: the force F¯ of any nature acting on a body of mass m leads to the 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 in the direction 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 meet with the movement of bodies along curvilinear trajectories. Consider how we can describe the acceleration in this case. For this, we assume that the velocity of a material point in the considered part of the trajectory can be written as:
v¯=vut¯
The speed v¯ is the product of its absolute value v byunit vector ut¯ directed along the tangent to the trajectory (tangential component).
According to the definition, acceleration is the derivative of speed with respect to time. We have:
a¯=dv¯/dt=d(vut¯)/dt=dv/dtut ¯ + vd(ut¯)/dt
The first term on the right side of the written equation is called tangential acceleration. Just like the velocity, it is directed along the tangent and characterizes the change in the absolute value v¯. The second term is the normal acceleration (centripetal), it is directed perpendicular to the tangent and characterizes the change in the magnitude vector v¯.
Thus, if the radius of curvature of the trajectory is equal to infinity (straight line), then the velocity vector does not change its direction in the process of moving the body. The latter means that the normal component of the total acceleration is zero.
In the case of a material point moving along a circle uniformly, the velocity modulus remains constant, that is, the tangential component of the total acceleration is equal to zero. The normal component is directed towards the center of the circle and is calculated by the formula:
a=v2/r
Here r is the radius. The reason for the appearance of centripetal acceleration is the action on the body of some internal force, which is directed towards the center of the circle. For example, for the movement of planets around the Sun, this force is gravitational attraction.
The formula that connects the full acceleration modules and itscomponent at(tangent), a (normal), looks like:
a=√(at2 + a2)
Uniformly accelerated movement in a straight line
Movement in a straight line with constant acceleration is often found in everyday life, for example, this is the movement of a car along the road. This kind of motion is described by the following velocity equation:
v=v0+ at
Here v0- some speed that the body had before its acceleration a.
If we plot the function v(t), we will get a straight line that intersects the y-axis at the point with coordinates (0; v0), and the tangent of the slope to the x-axis is equal to the acceleration modulus a.
Taking the integral of the function v(t), we get the formula for the path L:
L=v0t + at2/2
The graph of the function L(t) is the right branch of the parabola, which starts at the point (0; 0).
The above formulas are the basic equations of the kinematics of accelerated movement along a straight line.
If the body, having an initial speed v0, starts to slow down its movement with a constant acceleration, then we speak of uniformly slow movement. The following formulas are valid for it:
v=v0- at;
L=v0t - at2/2
Solving the problem of calculating acceleration
Being stillcondition, the vehicle starts moving. At the same time, in the first 20 seconds, he travels a distance of 200 meters. What is the acceleration of the car?
First, let's write down the general kinematic equation for the path L:
L=v0t + at2/2
Since in our case the vehicle was at rest, its speed v0 was equal to zero. We get the formula for acceleration:
L=at2/2=>
a=2L/t2
Substitute the value of the distance traveled L=200 m for the time interval t=20 s and write down the answer to the problem question: a=1 m/s2.
