The concept of full acceleration. acceleration components. Rapid motion in a straight line and uniform motion in a circle

2024 Author: Angel Austin | [email protected]. Last modified: 2023-12-17 05:14

When physics describes the movement of bodies, they use such quantities as force, speed, path of movement, angles of rotation, and so on. 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 fan 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 certain time. This acceleration in physics is called "acceleration". A more rigorous definition sounds like this: acceleration is a physical quantity that describes the speed or rate of change over time of the speed itself. Mathematically, this should be written as follows:

ā=dv¯/dt

Calculating the first time derivative of the speed, we will 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 writetotal average acceleration value ā_cp:

ā_cp=(v₂¯-v₁¯)/(t ₂-t₁).

This expression is similar to the previous one, only the body velocities are taken over a much longer period of time than dt.

The recorded formulas for the relationship 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 trajectory, then the acceleration is directed in the direction of the speed change.

Trajectory of motion and full acceleration vector

When studying the movement of bodies, special attention should be paid to the trajectory, that is, an imaginary line along which the movement occurs. In general, the trajectory is curvilinear. 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 obtain the formula for the total acceleration in terms of individual components, we represent the speed of the body at the point of the trajectory in the following form:

v¯=vu¯

Here u¯ is the unit vector tangent to the trajectory, v is the velocity model. Taking the time derivative of v¯ and simplifying the resulting terms, we arrive at the following equality:

ā=dv¯/dt=dv/dtu¯ + v²/rr_e¯.

The first term is the tangential acceleration componentā, the second term is the normal acceleration. Here r is the radius of curvature, r_e¯ is the unit length radius vector.

Thus, the total acceleration vector is the sum of mutually perpendicular vectors of tangential and normal acceleration, so 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 motion. The value of ā is always directed along the vector of the total force.

Mutual perpendicularity of the studied components a_t(tangential) and a (normal) allows us to write an expression for determining the total acceleration module:

a=√(a_t²+ a²⁾

Rectilinear rapid motion

If the trajectory is a straight line, then the velocity vector does not change during the motion of the body. This means that when describing the total 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 kinematic formulas of rectilinear uniformly accelerated or uniformly slow motion follow. Let's write them down:

v=v₀± at;

S=v₀t ± at²/2.

Here, the plus sign corresponds to accelerated movement, and the minus sign to slow movement (braking).

Uniform circular movement

Now let's look at how speed and acceleration are related in the case of body rotation around an axis. Let us assume that this rotation occurs at a constant angular velocity ω, that is, the body turns through equal angles in equal time intervals. Under the conditions described, the linear velocity v does not change its absolute value, but its vector is constantly changing. The last fact describes normal acceleration.

The formula for normal acceleration a has already been given above. Let's write it down again:

a=v²/r

This equality shows that, unlike the component a_t, the value a is not equal to zero even at a constant velocity modulus v. The larger this modulus, and the smaller the radius of curvature r, the greater the value of a . The appearance of normal acceleration is due to the action of the centripetal force, which tends to keep the rotating body on the circle line.