As you know, any physical quantity refers to one of two types, it is either scalar or vector. In this article, we consider such kinematic characteristics as speed and acceleration, and also show where the acceleration and velocity vectors are directed.
What is speed and acceleration?
Both values mentioned in this paragraph are important characteristics of any type of movement, whether it is the movement of a body in a straight line or along a curved path.
Speed is the speed of change of coordinates in time. Mathematically, this value is equal to the time derivative of the path traveled, that is:
v¯ = dl¯ / dt.
Here the vector l¯ is directed from the starting point of the path to the final.
In turn, acceleration is the speed with which speed itself changes in time. In the form of a formula, it can be written as follows:
a¯ = dv¯ / dt.
Obviously, taking the second derivative of the displacement vector l¯ in time, we also get the acceleration value.
Since the speed is measured in meters per second, the acceleration, according to the recorded expression, is measured in meters per second squared.
Where are the acceleration and velocity vectors directed?
In physics, it is customary to characterize any mechanical motion of a body with a certain trajectory. The latter is some imaginary curve along which the body moves in space. For example, a straight line or circle are vivid examples of common motion paths.
The body velocity vector is always directed in the direction of movement, regardless of whether the body slows down or accelerates, it moves in a straight line or in a curve. In geometric terms, the velocity vector is directed tangentially to the point of the trajectory at which the body is currently located.
The acceleration vector of a material or body point has nothing to do with speed. This vector is directed towards the change in speed. For example, for rectilinear motion, the quantity a¯ can either coincide in direction with v¯ or be opposite to v¯.
Body force and acceleration
We found that the acceleration vector of the body is directed towards the change in the velocity vector. Nevertheless, it is not always easy to determine how the velocity changes at a given point in the trajectory. Moreover, to determine the change in speed, it is necessary to perform the operation of the difference of vectors. To avoid these difficulties in determining the direction of the vector a¯, there is another way to quickly recognize it.
Below is the famous and well-known to every student Newton's law:
F¯ = m * a¯.
The formula shows that the cause of the acceleration in bodies is the force acting on them. Since the mass m is a scalar, the force vector F¯ and the acceleration vector a¯ are directed in the same way. This fact should be remembered and applied in practice whenever there is a need to determine the direction of a¯.
If several different forces act on the body, then the direction of the acceleration vector will be equal to the resulting vector of all forces.
Circular motion and acceleration
When the body moves in a straight line, the acceleration is directed either forward or backward. In the case of circular motion, the situation is complicated by the fact that the velocity vector is constantly changing its direction. In view of the foregoing, full acceleration is determined by its two components: tangential and normal accelerations.
Tangential acceleration is directed in exactly the same way as the velocity vector, or against it. In other words, this acceleration component is directed along the tangent to the trajectory. The tangential acceleration describes a change in the absolute value of the velocity itself.
The normal acceleration is directed along the normal to a given point of the trajectory, taking into account its curvature. In the case of circular motion, the vector of this component indicates the center, that is, the normal acceleration is directed along the radius of rotation. This component is often called centripetal.
Full acceleration is the sum of these components, so its vector can be directed arbitrarily with respect to the circle line.
If the body rotates without changing the linear velocity, then there is a non-zero only normal component, therefore the vector of full acceleration is directed to the center of the circle. Note that the force that holds the body along its path also acts on this center. For example, the gravitational force of the Sun holds our Earth and other planets in its orbits.