In order to be able to solve various problems on the motion of bodies in physics, it is necessary to know the definitions of physical quantities, as well as the formulas by which they are connected. This article will address issues of what is tangential velocity, what is full acceleration, and what components make up it.
Speed concept
The two main values of the kinematics of the movement of bodies in space are speed and acceleration. Speed describes the speed of movement, so the mathematical form of writing for it has the following form:
v¯ = dl¯ / dt.
Here l¯ - is the displacement vector. In other words, speed is the time derivative of the distance traveled.
As you know, every body moves along an imaginary line, which is called a trajectory. The velocity vector is always directed tangentially to this trajectory, no matter where the moving body is.
There are several names of the quantity v¯, if we consider it together with the trajectory. So, since it is directed along the tangent, it is called the tangential velocity. They can also be talked about as a linear physical quantity as opposed to angular velocity.
The speed in meters per second is calculated in SI, but in practice, kilometers per hour are often used.
The concept of acceleration
In contrast to the speed, which characterizes the speed of the body trajectory, acceleration is a value that describes the speed of change of speed, which is mathematically written as:
a¯ = dv¯ / dt.
Like speed, acceleration is a vector feature. However, its direction is not related to the velocity vector. It is determined by a change in direction v¯. If in the process of motion the velocity does not change its vector, then the acceleration a¯ will be directed along the same line as the velocity. Such acceleration is called tangential. If the speed will change direction, while maintaining the absolute value, then the acceleration will be directed to the center of curvature of the trajectory. It is called normal.
Acceleration is measured in m / s 2 . For example, the acceleration of gravity, known to everyone, is tangential when an object rises or falls vertically. Its value near the surface of our planet is 9.81 m / s 2 , that is, for every second of the fall, the speed of the body increases by 9.81 m / s.
The cause of acceleration is not speed, but strength. If the force F has an effect on a body of mass m, then it will inevitably create an acceleration a, which can be calculated as follows:
a = F / m.
This formula is a direct consequence of Newton’s second law.
Full, normal and tangential acceleration
Speed and acceleration as physical quantities were considered in the previous paragraphs. Now we will study in more detail which components make up the full acceleration a¯.
Suppose that a body moves with velocity v¯ along a curved path. Then the equality will be true:
v¯ = v * u¯.
The vector u¯ has unit length and is directed along the tangent line to the trajectory. Using this representation of the velocity v¯, we obtain the equality for full acceleration:
a¯ = dv¯ / dt = d (v * u¯) / dt = dv / dt * u¯ + v * du¯ / dt.
The first term obtained in the right equality is called tangential acceleration. The speed is associated with it by the fact that it quantitatively determines the change in the absolute value of v¯, without taking its direction into account.
The second term is normal acceleration. It quantitatively describes the change in the velocity vector, without taking into account the change in its modulus.
If we denote by a t and a n the tangential and normal components of the full acceleration a, then the modulus of the latter can be calculated by the formula:
a = √ (a t 2 + a n 2 ).
The relationship of tangential acceleration and speed
The corresponding relationship is described by kinematic expressions. For example, in the case of movement in a straight line with constant acceleration, which is tangential (the normal component is zero), the following expressions are true:
v = a t * t;
v = v 0 ± a t * t.
In the case of circular motion with constant acceleration, these formulas are also valid.
Thus, whatever the trajectory of the body’s movement, the tangential acceleration through the tangential velocity is calculated as the time derivative of its module, that is:
a t = dv / dt.
For example, if the speed changes according to the law v = 3 * t 3 + 4 * t, then a t will be equal to:
a t = dv / dt = 9 * t 2 + 4.
Speed and Normal Acceleration
We write in explicit form the formula for the normal component a n , we have:
a n ¯ = v * du¯ / dt = v * du¯ / dl * dl / dt = v 2 / r * r e ¯
Where r e ¯ is the unit length vector, which is directed to the center of curvature of the trajectory. This expression establishes the relationship between tangential velocity and normal acceleration. We see that the latter depends on the module v at a given time and on the radius of curvature r.
Normal acceleration always appears when the velocity vector changes; however, it is equal to zero if this vector preserves the direction. It makes sense to talk about the quantity a n ¯ only when the curvature of the trajectory is a finite quantity.
We noted above that when moving in a straight line, normal acceleration is absent. However, in nature there is a type of trajectory, when moving along which a n has a finite value, and a t = 0 for | v¯ | = const. This trajectory is a circle. For example, rotation with a constant frequency of a metal shaft, carousel or planet around its own axis occurs with constant normal acceleration a n and zero tangential acceleration a t .