When the motion of bodies is described in physics, then such quantities as force, speed, travel path, rotation angles and so on are used. This article will focus on one of the important quantities that combines the equations of kinematics and motion dynamics. Let's consider in detail what full acceleration is.
The concept of acceleration
Every lover of modern high-speed car brands knows that one of the important parameters for them is acceleration to a certain speed (usually up to 100 km / h) in a while. This acceleration in physics is called "acceleration". A more rigorous definition is: acceleration is a physical quantity that describes the speed or speed of a change in time of the speed itself. Mathematically, this should be written as follows:
ā = dv¯ / dt
After calculating the first time derivative of the velocity, we find the value of the instantaneous full acceleration ā.
If the movement is uniformly accelerated, then ā does not depend on time. This fact allows us to write the value of the total average acceleration ā cp :
ā cp = (v 2 ¯-v 1 ¯) / (t 2 -t 1 ).
This expression is similar to the previous one, only the values of the body velocities are taken over a much longer period of time than dt.
The written formulas for the relation between speed and acceleration allow us to draw a conclusion regarding the vectors of these quantities. If the speed is always directed tangentially to the motion path, then the acceleration is directed towards the change in speed.
Trajectory of motion and vector of full acceleration
When studying the motion of bodies, special attention should be paid to the trajectory, that is, the imaginary line along which the movement occurs. In the general case, the trajectory is curved. When moving along it, the speed of the body changes not only in magnitude, but also in direction. Since acceleration describes both components of the change in speed, it can be represented as the sum of two components. To get the formula for full acceleration through the individual components, imagine the speed of the body at the point of the trajectory in the following form:
v¯ = v * u¯
Here u¯ is the unit vector tangent to the trajectory, v is the velocity model. Taking the derivative of v¯ with respect to time and simplifying the obtained terms, we arrive at the following equality:
ā = dv¯ / dt = dv / dt * u¯ + v 2 / r * r e ¯.
The first term is the tangential component of the acceleration ā, the second term is the normal acceleration. Here r is the radius of curvature, r e ¯ is the unit length radius vector.
Thus, the full acceleration vector is the sum of mutually perpendicular vectors of tangential and normal acceleration, therefore its direction differs from the directions of the considered components and from the velocity vector.
Another way to determine the direction of the vector ā is to study the acting forces on the body in the process of its movement. The quantity ā is always directed along the vector of the total force.
The mutual perpendicularity of the studied components a t (tangential) and a n (normal) allows us to write an expression for determining the absolute acceleration modulus:
a = √ (a t 2 + a n 2 )
Rectilinear accelerated motion
If the trajectory is a straight line, then a change in the velocity vector in the process of body movement does not occur. This means that in describing full acceleration one should know only its tangential component a t . The normal component will be zero. Thus, the description of accelerated movement in a straight line is reduced to the formula:
a = a t = dv / dt.
From this expression all the kinematic formulas of a rectilinear uniformly accelerated or equally slow motion follow. We write them down:
v = v 0 ± a * t;
S = v 0 * t ± a * t 2/2.
Here, the plus sign corresponds to accelerated motion, and the minus sign corresponds to slow motion (braking).
Uniform movement around the circle
Now we will consider how speed and acceleration are connected in case of rotation of a body around an axis. Suppose that this rotation occurs at a constant angular velocity ω, that is, for equal intervals of time, the body rotates at equal angles. Under the described conditions, the linear velocity v does not change its absolute value, but its vector is constantly changing. The latter fact describes normal acceleration.
The formula for normal acceleration a n has already been given above. Let's write it again:
a n = v 2 / r
This equality shows that, in contrast to the component a t , the quantity a n is not equal to zero even with a constant modulus of speed v. The larger this modulus, and the smaller the radius of curvature r, the greater the value of a n . The appearance of normal acceleration is due to the action of a centripetal force, which tends to keep a rotating body on the circle line.