Speed is a physical quantity that characterizes the speed of movement and direction of movement of a material point relative to the chosen reference system; by definition, equal to the derivative of the radius vector of a point with respect to time.
Speed in a broad sense is the speed of change of any quantity (not necessarily the radius vector) depending on another (more often it means changes in time, but also in space or any other). So, for example, they talk about angular velocity, the rate of temperature change, the rate of a chemical reaction, the group velocity, the rate of connection, etc. Mathematically, the “rate of change” is characterized by the derivative of the quantity under consideration.
Acceleration is denoted by the rate of change of speed, that is, the first derivative of speed with respect to time, a vector quantity showing how much the velocity vector of a body changes as it moves per unit time:
acceleration is a vector, that is, it takes into account not only the change in the magnitude of the speed (the magnitude of the vector quantity), but also the change in its direction. In particular, the acceleration of a body moving in a circle with a constant absolute velocity is not zero; the body experiences a constant magnitude (and variable in direction) acceleration directed towards the center of the circle (centripetal acceleration).
The unit of acceleration in the International System of Units (SI) is meters per second per second (m/s2, m/s2),
The derivative of acceleration with respect to time, that is, the quantity characterizing the rate of change of acceleration, is called jerk:
Where is the jerk vector.
Acceleration is a quantity that characterizes the rate of change in speed.
Average acceleration
Average acceleration is the ratio of the change in speed to the period of time during which this change occurred. The average acceleration can be determined by the formula:
where is the acceleration vector.
The direction of the acceleration vector coincides with the direction of change in speed Δ = - 0 (here 0 is the initial speed, that is, the speed at which the body began to accelerate).
At time t1 (see Figure 1.8) the body has speed 0. At time t2 the body has speed . According to the rule of vector subtraction, we find the vector of change in speed Δ = - 0. Then the acceleration can be determined as follows:
The SI unit of acceleration is 1 meter per second per second (or meter per second squared), that is
A meter per second squared is equal to the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in one second. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m/s2, then this means that the speed of the body increases by 5 m/s every second.
Instant acceleration
The instantaneous acceleration of a body (material point) at a given moment of time is a physical quantity equal to the limit to which the average acceleration tends as the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:
The direction of acceleration also coincides with the direction of change in speed Δ for very small values of the time interval during which the change in speed occurs. The acceleration vector can be specified by projections onto the corresponding coordinate axes in a given reference system (projections aX, aY, aZ).
With accelerated linear motion, the speed of the body increases in absolute value, that is
and the direction of the acceleration vector coincides with the velocity vector 2.
If the speed of a body decreases in absolute value, that is
then the direction of the acceleration vector is opposite to the direction of the velocity vector 2. In other words, in this case the movement slows down, and the acceleration will be negative (and< 0). На рис. 1.9 показано направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.
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 normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.
Acceleration- a physical vector quantity that characterizes how quickly a body (material point) changes the speed of its movement. Acceleration is an important kinematic characteristic of a material point.
The simplest type of motion is uniform motion in a straight line, when the speed of the body is constant and the body covers the same path in any equal intervals of time.
But most movements are uneven. In some areas the body speed is greater, in others less. As the car begins to move, it moves faster and faster. and when stopping it slows down.
Acceleration characterizes the rate of change in speed. If, for example, the acceleration of a body is 5 m/s 2, then this means that for every second the speed of the body changes by 5 m/s, i.e. 5 times faster than with an acceleration of 1 m/s 2.
If the speed of a body during uneven motion changes equally over any equal periods of time, then the motion is called uniformly accelerated.
The SI unit of acceleration is the acceleration at which for every second the speed of the body changes by 1 m/s, i.e. meter per second per second. This unit is designated 1 m/s2 and is called “meter per second squared”.
Like speed, the acceleration of a body is characterized not only by its numerical value, but also by its direction. This means that acceleration is also a vector quantity. Therefore, in the pictures it is depicted as an arrow.
If the speed of a body during uniformly accelerated linear motion increases, then the acceleration is directed in the same direction as the speed (Fig. a); if the speed of the body decreases during a given movement, then the acceleration is directed in the opposite direction (Fig. b).
Average and instantaneous acceleration
The average acceleration of a material point over a certain period of time is the ratio of the change in its speed that occurred during this time to the duration of this interval:
\(\lt\vec a\gt = \dfrac (\Delta \vec v) (\Delta t) \)
The instantaneous acceleration of a material point at some point in time is the limit of its average acceleration at \(\Delta t \to 0\) . Keeping in mind the definition of the derivative of a function, instantaneous acceleration can be defined as the derivative of speed with respect to time:
\(\vec a = \dfrac (d\vec v) (dt) \)
Tangential and normal acceleration
If we write the speed as \(\vec v = v\hat \tau \) , where \(\hat \tau \) is the unit unit of the tangent to the trajectory of motion, then (in a two-dimensional coordinate system):
\(\vec a = \dfrac (d(v\hat \tau)) (dt) = \)
\(= \dfrac (dv) (dt) \hat \tau + \dfrac (d\hat \tau) (dt) v =\)
\(= \dfrac (dv) (dt) \hat \tau + \dfrac (d(\cos\theta\vec i + sin\theta \vec j)) (dt) v =\)
\(= \dfrac (dv) (dt) \hat \tau + (-sin\theta \dfrac (d\theta) (dt) \vec i + cos\theta \dfrac (d\theta) (dt) \vec j))v\)
\(= \dfrac (dv) (dt) \hat \tau + \dfrac (d\theta) (dt) v \hat n \),
where \(\theta \) is the angle between the velocity vector and the x-axis; \(\hat n \) - unit unit perpendicular to the speed.
Thus,
\(\vec a = \vec a_(\tau) + \vec a_n \),
Where \(\vec a_(\tau) = \dfrac (dv) (dt) \hat \tau \)- tangential acceleration, \(\vec a_n = \dfrac (d\theta) (dt) v \hat n \)- normal acceleration.
Considering that the velocity vector is directed tangent to the trajectory of motion, then \(\hat n \) is the unit unit of the normal to the trajectory of motion, which is directed to the center of curvature of the trajectory. Thus, normal acceleration is directed towards the center of curvature of the trajectory, while tangential acceleration is tangential to it. Tangential acceleration characterizes the rate of change in the magnitude of velocity, while normal acceleration characterizes the rate of change in its direction.
Movement along a curved trajectory at each moment of time can be represented as rotation around the center of curvature of the trajectory with angular velocity \(\omega = \dfrac v r\) , where r is the radius of curvature of the trajectory. In that case
\(a_(n) = \omega v = (\omega)^2 r = \dfrac (v^2) r \)
Acceleration measurement
Acceleration is measured in meters (divided) per second to the second power (m/s2). The magnitude of the acceleration determines how much the speed of a body will change per unit time if it constantly moves with such acceleration. For example, a body moving with an acceleration of 1 m/s 2 changes its speed by 1 m/s every second.
Acceleration units
- meter per second squared, m/s², SI derived unit
- centimeter per second squared, cm/s², derived unit of the GHS system
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Acceleration is rate of change of speed. In the SI system, acceleration is measured in meters per second squared (m/s 2), that is, it shows how much the speed of a body changes in one second.
If, for example, the acceleration of a body is 10 m/s 2 , then this means that for every second the speed of the body increases by 10 m/s. So, if before the start of acceleration the body was moving at a constant speed of 100 m/s, then after the first second of movement with acceleration its speed will be 110 m/s, after the second - 120 m/s, etc. In this case, the speed of the body gradually increased .
But the speed of the body can gradually decrease. This usually happens when braking. If the same body, moving at a constant speed of 100 m/s, begins to decrease its speed by 10 m/s every second, then after two seconds its speed will be 80 m/s. And after 10 seconds the body will stop altogether.
In the second case (when braking) we can say that the acceleration is negative. Indeed, to find the current speed after the start of braking, you need to subtract the acceleration multiplied by the time from the initial speed. For example, what is the speed of the body 6 seconds after braking? 100 m/s - 10 m/s 2 · 6 s = 40 m/s.
Since acceleration can take both positive and negative values, this means that acceleration is a vector quantity.
From the examples considered, we could say that when accelerating (increasing speed), acceleration is a positive value, and when braking, it is negative. However, everything is not so simple when we are dealing with a coordinate system. Here, speed also turns out to be a vector quantity, capable of being both positive and negative. Therefore, where the acceleration is directed depends on the direction of the speed, and not on whether the speed decreases or increases under the influence of the acceleration.
If the speed of the body is directed in the positive direction of the coordinate axis (say, X), then the body increases its coordinate for every second of time. So, if at the moment the measurement began, the body was at a point with a coordinate of 25 m and began to move at a constant speed of 5 m/s in the positive direction of the X axis, then after one second the body will be at a coordinate of 30 m, after 2 s - 35 m. In general, to find the coordinate of a body at a certain moment in time, you need to add the speed multiplied by the amount of elapsed time to the initial coordinate. For example, 25 m + 5 m/s · 7 s = 60 m. In this case, after 7 seconds the body will be at a point with coordinate 60. Here the speed is a positive value, since the coordinate increases.
Velocity is negative when its vector is directed in the negative direction of the coordinate axis. Let the body from the previous example begin to move not in the positive, but in the negative direction of the X axis at a constant speed. After 1 s the body will be at a point with a coordinate of 20 m, after 2 s - 15 m, etc. Now, to find the coordinate, you need to subtract the speed multiplied by the time from the initial one. For example, where will the body be in 8 s? 25 m - 5 m/s · 8 s = -15 m. That is, the body will be at a point with an x coordinate equal to -15. In the formula, we put a minus sign in front of the speed (-5 m/s), which means the speed is a negative value.
Let's call the first case (when the body moves in the positive direction of the X axis) A, and the second case B. Let's consider where the acceleration will be directed during braking and acceleration in both cases.
In case A, during acceleration, the acceleration will be directed in the same direction as the speed. Since the speed is positive, the acceleration will be positive.
In case A, when braking, the acceleration is directed in the opposite direction to the speed. Since the speed is a positive value, the acceleration will be negative, that is, the acceleration vector will be directed in the negative direction of the X axis.
In case B, during acceleration, the direction of acceleration will coincide with the direction of speed, which means the acceleration will be directed in the negative direction of the X axis (after all, speed is also directed there). Note that even though the acceleration is negative, it still increases the magnitude of the velocity.
In case B, when braking, the acceleration is in the opposite direction to the speed. Since the speed has a negative direction, the acceleration will be a positive value. But at the same time it will reduce the speed module. For example, the initial speed was -20 m/s, the acceleration was 2 m/s 2. The speed of the body after 3 s will be equal to -20 m/s + 2 m/s 2 · 3 s = -14 m/s.
Thus, the answer to the question “where is the acceleration directed” depends on what it is viewed in relation to. In relation to speed, acceleration can be directed in the same direction as the speed (during acceleration), or in the opposite direction (during braking).
In the coordinate system, positive and negative acceleration in itself does not say anything about whether the body was decelerating (reducing its speed) or accelerating (increasing its speed). We need to look at where the speed is directed.
1. Acceleration is a quantity that characterizes the change in speed per unit time. Knowing the acceleration of a body and its initial speed, you can find the speed of the body at any moment in time.
2. With any uneven movement, the speed changes. How does acceleration characterize this change?
2. If the acceleration of a body in magnitude is large, this means that the body quickly gains speed (when it accelerates) or quickly loses it (when braking).
3. How does “slow” linear motion differ from “accelerated” motion?
3. Movement with increasing absolute speed is called “accelerated” movement. Movement with decreasing speed in “slow” motion.
4. What is uniformly accelerated motion?
4. The motion of a body in which its speed changes equally over any period of time is called uniformly accelerated motion.
5. Can a body move at high speed but with low acceleration?
5. Maybe. Since acceleration does not depend on the value of speed, but characterizes only its change.
6. What is the direction of the acceleration vector during rectilinear uneven motion?
6. In case of rectilinear uneven motion, the acceleration vector a lies on the same straight line with the vectors V 0 and V .
7. Speed is a vector quantity, and both the magnitude of the speed and the direction of the speed vector can change. What exactly changes during rectilinear uniformly accelerated motion?
7. Speed module. Since the vectors V and a lie on the same line and the signs of their projections coincide.
Translational and rotational movements
Progressive is the movement of a rigid body in which any straight line drawn in this body moves while remaining parallel to its initial direction.
Translational motion should not be confused with rectilinear motion. When a body moves forward, the trajectories of its points can be any curved lines.
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- this is the ratio of the distance traveled to the time during which this path was traveled.
Speed is the same is the sum of the initial speed and acceleration multiplied by time.
Speed is the product of the angular velocity and the radius of the circle.
v=S/t
v=v 0 +a*t
v=ωR
Acceleration of a body during uniformly accelerated motion- a value equal to the ratio of the change in speed to the period of time during which this change occurred.
Tangential (tangential) acceleration– this is the component of the acceleration vector directed along the tangent to the trajectory at a given point of the movement trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.
Rice. 1.10. Tangential acceleration.
The direction of the tangential acceleration vector τ (see Fig. 1.10) coincides with the direction of linear velocity or is opposite to it. That is, the tangential acceleration vector lies on the same axis with the tangent circle, which is the trajectory of the body.
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 normal acceleration vector is perpendicular to the linear speed of movement (see Fig. 1.10). Normal acceleration characterizes the change in speed in direction and is denoted by the letter n. The normal acceleration vector is directed along the radius of curvature of the trajectory.
Full acceleration during curvilinear motion, it consists of tangential and normal accelerations along vector addition rule and is determined by the formula:
(according to the Pythagorean theorem for a rectangular rectangle).
The direction of total acceleration is also determined vector addition rule:
Angular velocity is a vector quantity equal to the first derivative of the angle of rotation of a body with respect to time:
v=ωR
Angular acceleration is a vector quantity equal to the first derivative of the angular velocity with respect to time:
Fig.3
When a 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 angular velocity. During accelerated motion, the vector ε codirectional to the vector ω (Fig. 3), when slowed down, it is opposite to it (Fig. 4).
Fig.4
Tangential component of acceleration a τ =dv/dt, v = ωR and
Normal component of acceleration
This means that the relationship between linear (path length s traversed by a point along a circular arc of radius R, linear velocity v, tangential acceleration a τ, normal acceleration a n) and angular quantities (rotation angle φ, angular velocity ω, angular acceleration ε) is expressed as follows formulas:
s = Rφ, v = Rω, and τ = R?, and n = ω 2 R.
In the case of uniform motion of a point along a circle (ω=const)
ω = ω 0 ± ?t, φ = ω 0 t ± ?t 2 /2,
where ω 0 is the initial angular velocity.
