Who discovered acceleration in physics. Uniformly accelerated motion, acceleration vector, direction, displacement

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 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 travels the same path in any equal intervals of time.

But most movements are uneven. In some areas, the speed of the body is greater, in others less. The car starts moving faster and faster. and when it stops, it slows down.

Acceleration characterizes the rate of change of speed. If, for example, the acceleration of the 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 the body during uneven movement for any equal intervals of time changes in the same way, then the movement is called uniformly accelerated.

The unit of acceleration in SI is such an 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, body acceleration is characterized not only by numerical value but also direction. This means that acceleration is also a vector quantity. Therefore, in the figures it is depicted as an arrow.

If the speed of the body during uniformly accelerated rectilinear motion increases, then the acceleration is directed in the same direction as the speed (Fig. a); if the speed of the body during this movement decreases, 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 \(\Delta t \to 0 \) . With the definition of the derivative of a function in mind, instantaneous acceleration can be defined as the time derivative of velocity:

\(\vec a = \dfrac (d\vec v) (dt) \)

Tangential and normal acceleration

If we write the speed as \(\vec v = v\hat \tau \) , where \(\hat \tau \) is the unit vector of the tangent to the motion trajectory, 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; \(\hat n \) - vector of the perpendicular to the velocity.

In this way,

\(\vec a = \vec a_(\tau) + \vec a_n \),

where \(\vec a_(\tau) = \dfrac (dv) (dt) \hat \tau \)- tangential acceleration, \(\vec a_n = \dfrac (d\theta) (dt) v \hat n \)- normal acceleration.

Given that the velocity vector is directed tangentially to the trajectory of motion, then \(\hat n \) is the vector of the normal to the trajectory of motion, which is directed towards 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 the speed, while normal characterizes the rate of change in its direction.

Movement along a curvilinear trajectory at each moment of time can be represented as a rotation around the center of curvature of the trajectory with an angular velocity \(\omega = \dfrac v r \) , where r is the radius of curvature of the trajectory. In this 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 the body will change per unit of time if it constantly moves with such an 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

• square meter per second, m/s², SI derived unit
• centimeter per second squared, cm/s², CGS derived unit

Acceleration characterizes the rate of change in the speed of a moving body. If the speed of a body remains constant, then it does not accelerate. 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 ​​of two speeds and time, or from the value of the force applied to the body.

Steps

Calculation of the average acceleration over two speeds

Formula for calculating the average acceleration. The average acceleration of a body is calculated from its initial and final speeds (speed is the speed of movement in a certain direction) and the time it takes for the body to reach the final speed. Formula for calculating acceleration: a = ∆v / ∆t, 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 n, where v to- final speed v n- starting speed, t to- end time t n- start time.

• Since acceleration has a direction, always subtract initial speed from final speed; otherwise, the direction of the calculated acceleration will be wrong.
• If the initial time is not given in the problem, then it is assumed that t n = 0.

1. Find the acceleration using the formula. First, write the formula and the variables given to you. Formula: . Subtract the initial speed from the final speed, and then divide the result by the time span (change in time). You will get the average acceleration for a given period of time.

   • If the final speed is less than the initial one, then the acceleration is negative meaning, that is, the body slows down.
   • 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 to - v n) / (t to - t n)
     • Write variables: v to= 46.1 m/s, v n= 18.5 m/s, t to= 2.47 s, t n= 0 s.
     • Calculation: a\u003d (46.1 - 18.5) / 2.47 \u003d 11.17 m / s 2.
   • Example 2: A motorcycle starts braking at 22.4 m/s and stops after 2.55 seconds. Find the average acceleration.
     • Write the formula: a \u003d Δv / Δt \u003d (v to - v n) / (t to - t n)
     • Write variables: v to= 0 m/s, v n= 22.4 m/s, t to= 2.55 s, t n= 0 s.
     • Calculation: a\u003d (0 - 22.4) / 2.55 \u003d -8.78 m / s 2.

Force Acceleration Calculation

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. Such acceleration depends on the resultant 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 = m x a, where F res is the resultant force acting on the body, m- body mass, a is the acceleration of the body.
   • When working with this formula, use the units of the metric system, 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 the mass of the body. To do this, put the body on the scales and find its mass in grams. If you are looking at a very large body, look up its mass in reference books or on the Internet. The mass of large bodies is measured in kilograms.

   • To calculate the acceleration using the above formula, you must convert grams to kilograms. Divide the mass in grams by 1000 to get the mass in kilograms.

3. Find the resultant force acting on the body. The resulting force is not balanced by other forces. If two oppositely directed forces act on a 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. Acceleration occurs when a force acts on a body that is not balanced by other forces and which leads to a change in the speed of the body in the direction of this force.

   Transform the formula F = ma to calculate the acceleration. To do this, divide both sides of this formula by m (mass) and get: a = F / m. Thus, to find the acceleration, divide the force by the mass of the accelerating body.

   • The force is directly proportional to the acceleration, that is, more power acting on the body, the faster it accelerates.
   • Mass is inversely proportional to acceleration, that is, the greater the mass of the body, the slower it accelerates.

4. Calculate the acceleration using the resulting formula. Acceleration is equal to the quotient of the resultant force acting on the body divided by its mass. Substitute the values ​​given to you into this formula to calculate the body's acceleration.

   • For example: a force equal to 10 N acts on a body of mass 2 kg. Find the acceleration of the body.
   • a = F/m = 10/2 = 5 m/s 2

Testing your knowledge

1. direction of acceleration. The scientific concept of acceleration does not always coincide with the use of this quantity in Everyday life. Remember that acceleration has a direction; acceleration has positive value, if it is directed up or to the right; acceleration has a negative value if it is directed downwards or to the left. Check the correctness of your solution based on the following table:

2. Example: a toy boat with a mass of 10 kg is moving north with an acceleration of 2 m/s 2 . wind blowing in westbound, acts on the boat with a force of 100 N. Find the acceleration of the boat in the north direction.
3. Solution: Since the force is perpendicular to the direction of motion, it does not affect the motion in that direction. Therefore, the acceleration of the boat in the northern direction will not change and will be equal to 2 m / s 2.

4. resultant force. If several forces act on the body at once, find the resulting force, and then proceed to calculate the acceleration. Consider the following problem (in two dimensions):

   • Vladimir pulls (on the right) a 400 kg container with a force of 150 N. Dmitry pushes (on the left) a container with a force of 200 N. The wind blows from right to left and acts on the container with a force of 10 N. Find the acceleration of the container.
   • Solution: The condition of this problem is designed to confuse you. In fact, everything is very simple. Draw a diagram of the direction of forces, so you will see that a force of 150 N is directed to the right, a force of 200 N is also directed to the right, but a force of 10 N is directed to the left. Thus, the resulting force is: 150 + 200 - 10 = 340 N. The acceleration is: a = F / m = 340/400 = 0.85 m / s 2.

Let us consider in more detail what is acceleration in physics? This is a message to the body of additional speed per unit of time. AT international system units (SI) per unit of acceleration is considered to be the number of meters traveled per second (m/s). For the off-system unit Gal (Gal), which is used in gravimetry, the acceleration is 1 cm/s 2 .

Types of accelerations

What is acceleration in formulas. The type of acceleration depends on the motion vector of the body. In physics, this can be movement in a straight line, along a curved line, and along a circle.

1. If an object moves in a straight line, the motion will be uniformly accelerated, and linear accelerations will begin to act on it. The formula for calculating it (see formula 1 in Fig.): a=dv/dt
2. If we are talking about the motion of a body in a circle, then the acceleration will consist of two parts (a=a t +a n): tangential and normal acceleration. Both of them are characterized by the speed of movement of the object. Tangential - by changing the speed modulo. Its direction is tangent to the path. Such an acceleration is calculated by the formula (see formula 2 in Figure): a t =d|v|/dt
3. If the speed of an object moving along a circle is constant, the acceleration is called centripetal or normal. The vector of such an acceleration is constantly directed towards the center of the circle, and the value of the module is (see formula 3 in Fig.): |a(vector)|=w 2 r=V 2 /r
4. When the speed of the body around the circumference is different, there is an angular acceleration. It shows how it has changed angular velocity per unit of time and is equal to (see formula 4 in Fig.): E (vector) \u003d dw (vector) / dt
5. In physics, options are also considered when the body moves in a circle, but at the same time approaches or moves away from the center. In this case, Coriolis accelerations act on the object. When the body moves along a curved line, its acceleration vector will be calculated by the formula (see formula 5 in Figure): a (vector)=a T T+a n n(vector)+a b b(vector) =dv/dtT+v 2 /Rn(vector)+a b b(vector), in which:

• v - speed
• T (vector) - unit vector tangent to the trajectory, going along the velocity (tangent unit vector)
• n (vector) - the vector of the principal normal with respect to the trajectory, which is defined as a unit vector in the direction dT (vector)/dl
• b (vector) - ort of the binormal with respect to the trajectory
• R - radius of curvature of the trajectory

In this case, the binormal acceleration a b b (vector) is always equal to zero. Therefore, the final formula looks like this (see formula 6 in Figure): a (vector)=a T T+a n n(vector)+a b b(vector)=dv/dtT+v 2 /Rn(vector)

What is free fall acceleration?

acceleration free fall(denoted by the letter g) is called the acceleration that is given to an object in a vacuum by gravity. According to Newton's second law, this acceleration is equal to the force of gravity acting on an object of unit mass.

On the surface of our planet, the value of g is usually called 9.80665 or 10 m / s². To calculate the real g on the surface of the Earth, some factors will need to be taken into account. For example, latitude and time of day. So the value of true g can be from 9.780 m/s² to 9.832 m/s² at the poles. To calculate it, an empirical formula is used (see formula 7 in Fig.), in which φ is the latitude of the area, and h is the distance above sea level, expressed in meters.

Formula for calculating g

The fact is that such acceleration of free fall consists of gravitational and centrifugal acceleration. The approximate value of the gravitational one can be calculated by representing the Earth as a homogeneous ball with mass M, and calculating the acceleration along its radius R (formula 8 in Fig. .

If we use this formula to calculate the gravitational acceleration on the surface of our planet (mass M = 5.9736 10 24 kg, radius R = 6.371 10 6 m), formula 9 in Fig. 9 will be obtained, however, this value conditionally coincides with what is speed, acceleration specific location. The discrepancies are due to several factors:

• Centrifugal acceleration taking place in the reference frame of the planet's rotation
• The fact that the planet Earth is not spherical
• The fact that our planet is heterogeneous

Instruments for measuring acceleration

Acceleration is usually measured with an accelerometer. But he calculates not the acceleration itself, but the reaction force of the support that occurs during accelerated movement. The same resistance forces appear in the gravitational field, so gravity can also be measured with an accelerometer.

There is another device for measuring acceleration - an accelerograph. It calculates and graphically captures the acceleration values ​​of translational and rotational motion.

Acceleration is a value that characterizes the rate of change of speed.

For example, a car, moving away, increases the speed of movement, that is, it moves at an accelerated pace. Initially, its speed is zero. Starting from a standstill, the car gradually accelerates to a certain speed. If a red traffic light lights up on its way, the car will stop. But it will not stop immediately, but after some time. That is, its speed will decrease down to zero - the car will move slowly until it stops completely. However, in physics there is no term "deceleration". If the body is moving, slowing down, then this will also be the acceleration of the body, only with a minus sign (as you remember, speed is a vector quantity).

> is the ratio of the change in speed to the time interval during which this change occurred. The average acceleration can be determined by the formula:

Rice. 1.8. Average acceleration. in 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 point moving in a straight line, at which in one second the speed of this point increases by 1 m / s. In other words, acceleration determines how much the speed of a body changes in one second. For example, if the acceleration is 5 m / s 2, then this means that the speed of the body increases by 5 m / s every second.

Instantaneous acceleration of a body (material point) in this moment time is physical quantity, equal to the limit to which the average acceleration tends when the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:

With accelerated rectilinear motion, the speed of the body increases in absolute value, that is

V2 > v1

and the direction of the acceleration vector coincides with the velocity vector

If the modulo velocity of the body decreases, that is

V 2< v 1

then the direction of the acceleration vector is opposite to the direction of the velocity vector In other words, in this case, deceleration, while the acceleration will be negative (and< 0). На рис. 1.9 показано направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

Rice. 1.9. Instant acceleration.

When moving along a curvilinear trajectory, not only the modulus of speed changes, but also its direction. In this case, the acceleration vector is represented as two components (see the next section).

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

Rice. 1.10. tangential acceleration.

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

Normal acceleration

Normal acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. 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 the direction and is denoted by the letter The vector of normal acceleration is directed along the radius of curvature of the trajectory.

Full acceleration

Full acceleration in curvilinear motion, it consists of tangential and normal acceleration by and is determined by the formula:

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