Acceleration definition. Speed

Displacement (in kinematics) - a change in the location of a physical body in space relative to the selected frame of reference. Also called displacement is a vector characterizing this change. Possesses the property of additivity.

Velocity (often denoted from the English velocity or the French vitesse) is a vector physical quantity that characterizes the speed of movement and direction of movement of a material point in space relative to the selected reference system (for example, angular velocity).

Acceleration (usually denoted in theoretical mechanics) is the derivative of velocity with respect to time, a vector quantity that shows how much the velocity vector of a point (body) changes as it moves per unit of time (i.e. acceleration takes into account not only the change in the magnitude of the velocity, but also its directions).

Tangential (tangential) acceleration Is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the trajectory of motion. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.

Figure: 1.10. Tangential acceleration.

The direction of the vector of tangential acceleration τ (see Fig. 1.10) coincides with the direction of the linear velocity or opposite to it. That is, the vector of tangential acceleration lies on the same axis with the tangent circle, which is the trajectory of the body.

Normal acceleration

Normal acceleration Is the component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body. That is, the vector of normal acceleration is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration is the directional change in speed and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.

Full acceleration

Full acceleration in curvilinear motion, it is composed of tangential and normal accelerations according to the vector addition rule and is determined by the formula:

(according to the Pythagorean theorem for a rectangular rectangle).

The direction of full acceleration is also determined by the vector addition rule:

Force. Weight. Newton's laws.

Sila is a vector physical quantity that is a measure of the intensity of the impact on a given body of other bodies, as well as fields. The force applied to a massive body is the cause of a change in its speed or the appearance of deformations in it.

Mass (from the Greek μάζα) is a scalar physical quantity, one of the most important quantities in physics. Initially (XVII-XIX centuries), it characterized the "amount of matter" in a physical object, on which, according to the ideas of that time, depended both on the object's ability to resist the applied force (inertia) and gravitational properties - weight. It is closely related to the concepts of “energy” and “impulse” (according to modern concepts, mass is equivalent to rest energy).

Newton's first law

There are such frames of reference, called inertial, relative to which a material point, in the absence of external influences, retains the magnitude and direction of its speed for an unlimited time.

Newton's second law

In the inertial reference system, the acceleration that a material point receives is directly proportional to the resultant of all forces applied to it and inversely proportional to its mass.

Newton's third law

Material points act in pairs on each other with forces of the same nature, directed along the straight line connecting these points, equal in magnitude and opposite in direction:

Pulse. Impulse conservation law. Elastic and inelastic impacts.

Impulse (Momentum) is a vector physical quantity that characterizes the measure of the mechanical movement of the body. In classical mechanics, the momentum of a body is equal to the product of the mass m of this body by its velocity v, the direction of the momentum coincides with the direction of the velocity vector:

The law of conservation of momentum (the Law of conservation of momentum) states that the vector sum of the momenta of all bodies (or particles) of a closed system is a constant value.

In classical mechanics, the momentum conservation law is usually derived as a consequence of Newton's laws. From Newton's laws, it can be shown that when moving in empty space, the momentum is conserved in time, and in the presence of interaction, the rate of its change is determined by the sum of the applied forces.

Like any of the fundamental conservation laws, the momentum conservation law describes one of the fundamental symmetries, the homogeneity of space.

Absolutely inelastic blow is called such an impact interaction in which the bodies are connected (stick together) with each other and move on as one body.

With a completely inelastic impact, mechanical energy is not conserved. It partially or completely passes into the internal energy of bodies (heating).

Absolutely resilient impact a collision is called, in which the mechanical energy of a system of bodies is conserved.

In many cases, collisions of atoms, molecules and elementary particles obey the laws of absolutely elastic impact.

With an absolutely elastic impact, along with the law of conservation of momentum, the law of conservation of mechanical energy is fulfilled.

4. Types of mechanical energy. Job. Power. Law of energy conservation.

In mechanics, two types of energy are distinguished: kinetic and potential.

Kinetic energy is the mechanical energy of any freely moving body and is measured by the work that the body could perform when it was decelerated to a complete stop.

So, the kinetic energy of a translationally moving body is equal to half the product of the mass of this body by the square of its speed:

Potential energy is the mechanical energy of a system of bodies, determined by their mutual arrangement and the nature of the forces of interaction between them. Numerically, the potential energy of the system in its given position is equal to the work that will be performed by the forces acting on the system when the system moves from this position to where the potential energy is conventionally assumed to be zero (E n \u003d 0). The concept of "potential energy" takes place only for conservative systems, i.e. systems in which the work of the acting forces depends only on the initial and final position of the system.

So, for a load of weight P, lifted to a height h, the potential energy will be equal to E n \u003d Ph (E n \u003d 0 at h \u003d 0); for a load attached to a spring, E n \u003d kΔl 2/2, where Δl is the elongation (compression) of the spring, k is its stiffness coefficient (E n \u003d 0 at l \u003d 0); for two particles with masses m 1 and m 2, which are attracted by the law of universal gravitation, , where γ is the gravitational constant, r is the distance between particles (E n \u003d 0 as r → ∞).

The term "work" in mechanics has two meanings: work as a process in which a force moves a body, acting at an angle other than 90 °; work is a physical quantity equal to the product of force, displacement and cosine of the angle between the direction of the force and displacement:

Work is zero when the body moves by inertia (F \u003d 0), when there is no movement (s \u003d 0), or when the angle between movement and force is 90 ° (cos a \u003d 0). The unit of work in SI is the joule (J).

1 joule is such work that is performed by a force of 1 N when the body moves 1 m along the line of action of the force. To determine the speed of the work, enter the "power" value.

Power is a physical quantity equal to the ratio of the work performed over a certain period of time to this period of time.

Distinguish the average power over a period of time:

and instantaneous power at a given time:

Since work is a measure of energy change, power can also be defined as the rate of change in system energy.

In the SI system, the unit of measure for power is the watt, which is equal to one joule divided by a second.

The law of conservation of energy is a fundamental law of nature, established empirically and consisting in the fact that for an isolated physical system, a scalar physical quantity can be introduced, which is a function of the parameters of the system and is called energy, which is conserved over time. Since the law of conservation of energy does not refer to specific quantities and phenomena, but reflects a general law that is applicable everywhere and always, then it can be called not a law, but the principle of conservation of energy.

Acceleration characterizes the rate of change in the speed of a moving body. If the speed of the body remains constant, then it is not accelerating. Acceleration takes place only when the speed of the body changes. If the speed of a body increases or decreases by some constant value, then such a body moves with constant acceleration. Acceleration is measured in meters per second per second (m / s 2) and is calculated from the values \u200b\u200bof two speeds and time, or from the value of the force applied to the body.

Steps

Calculating the average acceleration over two speeds

Formula for calculating average acceleration. The average acceleration of a body is calculated from its initial and final velocities (velocity is the speed at which it moves in a certain direction) and the time it takes for the body to reach its final velocity. Formula for calculating acceleration: a \u003d Δv / Δt, where a is the acceleration, Δv is the change in speed, Δt is the time required to reach the final speed.

Definition of variables. You can calculate Δv and Δt in the following way: Δv \u003d v to - v n and Δt \u003d t to - t nwhere v to - final speed, v n - starting speed, t to - end time, t n - start time.

Since acceleration has a direction, always subtract the start speed from the end speed; otherwise, the direction of the calculated acceleration will be incorrect.
If the initial time is not given in the problem, then it is assumed that t n \u003d 0.

Find acceleration using the formula. First, write the formula and the variables given to you. Formula: ... Subtract the start speed from the end speed, and then divide the result by the amount of time (change in time). You will get the average acceleration over a given period of time.
- If the final speed is less than the initial one, then the acceleration has a negative value, that is, the body decelerates.
- Example 1: A car accelerates from 18.5 m / s to 46.1 m / s in 2.47 s. Find the average acceleration.
  - Write the formula: a \u003d Δv / Δt \u003d (v k - v n) / (t k - t n)
  - Write the variables: v to \u003d 46.1 m / s, v n \u003d 18.5 m / s, t to \u003d 2.47 s, t n \u003d 0 s.
  - Calculation: a \u003d (46.1 - 18.5) / 2.47 \u003d 11.17 m / s 2.
- Example 2: The motorcycle starts decelerating at 22.4 m / s and stops after 2.55 s. Find the average acceleration.
  - Write the formula: a \u003d Δv / Δt \u003d (v k - v n) / (t k - t n)
  - Write the variables: v to \u003d 0 m / s, v n \u003d 22.4 m / s, t to \u003d 2.55 s, t n \u003d 0 s.
  - Calculation: and \u003d (0 - 22.4) / 2.55 \u003d -8.78 m / s 2.
Calculating force acceleration
1. Newton's second law. According to Newton's second law, a body will accelerate if the forces acting on it do not balance each other. This acceleration depends on the resulting force acting on the body. Using Newton's second law, you can find the acceleration of a body if you know its mass and the force acting on that body.
  - Newton's second law is described by the formula: F res \u003d m x awhere F res - the resulting force acting on the body, m - body mass, a - body acceleration.
  - When working with this formula, use the metric units of measurement, in which mass is measured in kilograms (kg), force in newtons (N), and acceleration in meters per second per second (m / s 2).
2. Find your body weight. To do this, put the body on a scale and find its mass in grams. If you are considering a very large body, look for its mass in reference books or on the internet. The mass of large bodies is measured in kilograms.
  - To calculate acceleration using the above formula, you need to convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.
3. Find the resulting force on the body. The resulting force is not counterbalanced by other forces. If two oppositely directed forces act on the body, and one of them is greater than the other, then the direction of the resulting force coincides with the direction of the greater force.

For example, a car that starts from a standstill moves at an accelerated rate, as it increases its speed. At the starting point, the vehicle speed is zero. Having started moving, the car accelerates to a certain speed. If it is necessary to brake, the car will not be able to stop instantly, but for some time. That is, the speed of the car will tend to zero - the car will start moving slowly until it stops completely. But physics has no term "deceleration". If the body moves, reducing its speed, this process is also called acceleration , but with a "-" sign.

Average acceleration called the ratio of the change in speed to the time interval for which this change occurred. Calculate the average acceleration using the formula:

where is it . The direction of the acceleration vector is the same as that of the direction of speed change Δ \u003d - 0

where 0 is the starting speed. At a moment in time t 1 (see the figure below) at body 0. At a moment in time t 2 the body has speed. Based on the rule of subtraction of vectors, we determine the vector of change in velocity Δ \u003d - 0. From here we calculate the acceleration:

SI unit of acceleration called 1 meter per second per second (or meter per second squared):

A meter per second squared is the acceleration of a rectilinear moving point, at which in 1 s the speed of this point increases by 1 m / s. In other words, acceleration determines the rate of change in the body's velocity in 1 s. For example, if the acceleration is 5 m / s 2, then the speed of the body increases by 5 m / s every second.

Instant acceleration of a body (material point) at a given moment of time is a physical quantity that is equal to the limit to which the average acceleration tends when the time interval tends to 0. In other words, this is the acceleration developed by the body in a very small period of time:

Acceleration has the same direction as the change in speed Δ in extremely small time intervals during which the speed changes. The acceleration vector can be specified using projections on the corresponding coordinate axes in a given reference system (projections a X, a Y, a Z).

With accelerated rectilinear motion, the speed of the body increases in magnitude, i.e. v 2\u003e v 1, and the acceleration vector has the same direction as the velocity vector 2.

If the speed of the body decreases in absolute value (v 2< v 1), значит, у вектора ускорения направление противоположно направлению вектора скорости 2 . Другими словами, в таком случае наблюдаем slowing down (acceleration is negative, and< 0). На рисунке ниже изображено направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

If there is movement along a curved trajectory, then the modulus and direction of speed change. This means that the acceleration vector is represented in the form of 2 components.

Tangential (tangential) accelerationis called that component of the acceleration vector, which is directed tangentially to the trajectory at a given point of the trajectory of motion. Tangential acceleration describes the degree of change in speed modulo when making a curvilinear motion.

Have vector of tangential acceleration τ (see figure above) the direction is the same as that of the linear velocity or opposite to it. Those. the vector of tangential acceleration is in the same axis with the tangent circle, which is the trajectory of the body.

Translational and rotational movements

Translationala motion of a rigid body is called such that any straight line drawn in this body moves, remaining parallel to its initial direction.

The translational motion should not be confused with the rectilinear motion. During the translational motion of the body, the trajectories of its points can be any curved lines.

The rotational motion of a rigid body around a fixed axis is such a motion in which any two points belonging to the body (or invariably associated with it) remain motionless throughout the movement

Speed is the ratio of the distance traveled to the time during which this route has been covered.
The speed is the same is the sum of initial speed and acceleration times time.
Speed - the product of the angular velocity and the radius of the circle.

v \u003d S / t
v \u003d v 0 + a * t
v \u003d ωR

Acceleration of the body, with uniformly accelerated motion - a value equal to the ratio of the change in speed to the time interval during which this change occurred.

Figure: 1.10. Tangential acceleration.

Full acceleration in curvilinear motion, it is made up of tangential and normal accelerations along vector addition rule and is determined by the formula:

(according to the Pythagorean theorem for a rectangular rectangle).

Full acceleration direction is also determined vector addition rule:

Angular velocity is called a vector quantity equal to the first derivative of the angle of rotation of the body with respect to time:

v=ωR

Angular acceleration is called a vector quantity equal to the first derivative of the angular velocity with respect to time:

Fig. 3

When the body rotates around a fixed axis, the angular acceleration vector ε directed along the axis of rotation towards the vector of the elementary increment of the angular velocity. With accelerated motion, the vector ε co-directional with vector ω (fig. 3), at slow motion - opposite to it (fig. 4).

Fig. 4

The tangential component of acceleration a τ \u003d dv / dt, v \u003d ωR and

Normal component of acceleration

This means that the relationship between linear (the length of the path s traversed by a point along an arc of a circle of radius R, linear velocity v, tangential acceleration a τ, normal acceleration a n) and angular quantities (angle of rotation φ, angular velocity ω, angular acceleration ε) is expressed as follows formulas:

s \u003d Rφ, v \u003d Rω, and τ \u003d R ?, a n \u003d ω 2 R.
In the case of an equally variable motion of a point along a circle (ω \u003d const)

ω \u003d ω 0 ±? t, φ \u003d ω 0 t ±? t 2/2,
where ω 0 is the initial angular velocity.

And why is it needed. We already know what a frame of reference, relativity of motion and a material point are. Well, it's time to move on! Here we will look at the basic concepts of kinematics, put together the most useful formulas for the basics of kinematics, and give a practical example of solving the problem.

Let's solve the following problem: the point moves in a circle with a radius of 4 meters. The law of its motion is expressed by the equation S \u003d A + Bt ^ 2. A \u003d 8m, B \u003d -2m / s ^ 2. At what point in time is the normal acceleration of a point equal to 9 m / s ^ 2? Find the speed, tangential and total acceleration of a point for this moment in time.

Solution: we know that in order to find the speed, we need to take the first time derivative of the law of motion, and the normal acceleration is equal to the quotient of the square of the speed and the radius of the circle along which the point is moving. Armed with this knowledge, we will find the required values.

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