Measurement of the acceleration

Measurement of the acceleration

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Are you struggling to understand the Lesson Measurement of Acceleration? Then you may be missing the following basics:

Acceleration45 min.


In this learning section, the concept of acceleration is introduced and explained using examples. Movements in different coordinate and reference systems are considered.

Overview of the sensors20 min.

ChemistryPhysical chemistrySensors

Introduction to sensors; Explanation of the basic terms

  • What is the complete sentence called? When accelerating changes in one. -Interval the.
  • What do you need to be able to measure accelerations? Tip: There is something like this in smartphones and tablets, for example.
  • With most acceleration processes, the speed changes very quickly. What is the name of the acceleration that occurs here and what acceleration is generally expected in physics?
  • How do you calculate a speed difference and what abbreviation is used for it in physics?

Below you will find the answers to the questions and more detailed information.


The acceleration ( vec a ) is the temporal change or derivation of the speed ( vec v ) of a body in terms of magnitude and / or direction.

If the amount of speed decreases, d. H. in the case of negative acceleration, one also speaks of Brakes. At a constant acceleration there is a evenly accelerated movement before. In the case of an acceleration that varies over time, the quotient ( dfrac < Delta vec v> < Delta t> ) is called Average acceleration and the value ( vec a (t) ) as the instantaneous acceleration at the time t.

The acceleration at any point on the trajectory of a mass point can be converted into a tangential component (Tangential acceleration, Orbit acceleration ( vec a_ text t )) and a component perpendicular (Centripetal acceleration ( vec a_ text)) to form a trajectory. ( vec a_ text t ) is decisive for the change in the amount of speed, ( vec a_ text) for changing direction.

In the case of rotary movements, the angular acceleration is the analogue of the acceleration.

The connection between the acceleration of a body and the force acting on it, Newton's second axiom, is one of the most important laws of physics.


Dear Mr. Richard P.,

thanks for the hint. We have now left a timeline comment explaining the unit m / s² on the video.

Greetings from the editorial team

Explanation not bad, but m / s2 should be explained - why s2.
otherwise there is no motivation - one could bring the topic more "explanatory" instead of just "read it down"! Such videos tend to "scare away" the children.

Maybe next time bring a little more joy with you, otherwise you will fall asleep.

A bit dull and boring but explained well

Try now for free 89,999 videos, exercises and Worksheets!

Measurement of acceleration - chemistry and physics

Acceleration measures how quickly the speed of a body changes per unit of time. Analogous to the speed, the speed change per time unit (i.e. per s,.) Is measured for a sufficiently small time interval & # 916t.

It's easy with linear (one-dimensional) movements: Here it is only about changes of the one velocity coordinate v (which can be positive and negative). Look at this in a t-v diagram:

A time interval of length & # 916t is considered. The speed coordinate (the "speed") v changes by & # 916v.

The quotient & # 916v / & # 916t (speed change per unit of time) is then just the slope of the t-v diagram.

The acceleration a (acceleration coordinate) is interpreted as the slope of the t-v diagram:

This also results in a measurement method. The interpretation of a as the slope of the t-v diagram is extremely useful for many considerations. You only need to determine whether the t-v graph rises or falls you already know the sign of the acceleration (positive or negative).

Positive acceleration a means: the speed increases, negative acceleration means: the speed decreases. But that only has something to do indirectly with speeding up or slowing down:

In the entire area under consideration, the speed decreases with increasing time. The speed is highest at time 0, and lowest at time 4 s. Accordingly, the acceleration is negative. But from 0 s to 2 s the body gets slower and slower until it finally stops at 2 s. Then the body speeds up again, moving in the opposite direction. At 4 s the body is just as fast as at 0 s, but it moves in the opposite direction.

Measurement of acceleration a:

The signs of & # 916v and & # 916t result when you go from A to B. The path opposite to the coordinate direction is followed by a negative sign, while the coordinate direction is followed by a positive sign.

So & # 916v is negative (-20 m / s), & # 916t is positive (4 s). The measurement of a is shown in the picture.

Always assume the following sense of passage:

The hand points to the right towards the fingertips:

According to the definition, a unit of acceleration could be a = & # 916v / & # 916t: [a] = 1 km / h / s = 1 km / (h s). That would make sense if the speed were to change by 1 km / h in 1 s.

The usual unit, however, is [a] = 1 m / s / s = 1 m / s 2, suitable for a process in which a speed change of 1 m / s takes place in 1 s.

In general, one has to study vectors:

A time interval of length & # 916t is considered. At the beginning is the speed v0, at the end v. So she has to & # 916v = v - v0 changed.

The vectors of speed change and acceleration, & # 916v and a, are directed the same way!

The acceleration vector a is then defined as

Acceleration is a change in speed & # 916v per time interval & # 916t.

To do this, you choose a sufficiently small time interval so that you do not also change the acceleration vector a must consider in this interval.

You can also see it differently: & # 916v = a & # 916t is a too a rectified vector. The acceleration vector a has the same direction as the vector of the speed change & # 916v.

In the simplest circular motion, the amount of the speed remains constant, but the direction of the speed vector v Constantly changing.

A speed change occurs & # 916v, Sequence of an acceleration vector a, which for a very small time interval & # 916t is rectified to & # 916v is (& # 916v = a & # 916t), namely towards the center of the circle!

Later you will also learn that force (vector) F. and acceleration (vector) a are rectified. The acceleration a always has the same direction as the force F.. It is the total force.

Maybe you say of a fast moving body, he "is accelerated ". You're right: sometime in the past is it probably accelerates in the physical sense beenso he can move quickly now. In physics, however, the term acceleration is used differently. They say "a body will accelerates ", meaning that the body, while it is being viewed, its speed Changes, no matter what happened in the past. If you continue claim of a fast moving body that he "is accelerated, "your teacher will probably protest violently.

Table of contents

In order to calculate the acceleration for a certain point in time instead of for a time interval, one has to switch from the difference quotient to the differential quotient. The acceleration is then the first time derivative of the speed with respect to time:

Since the velocity is the derivative of the position with respect to time, the acceleration can also be used as the second derivative of the position vector r → < displaystyle < vec >> represent:

The time derivative of the acceleration (i.e. the third derivative of the position vector with respect to time) becomes Jerk j → < displaystyle < vec >> called:

Examples of calculation using the Edit speed

The speed has increased by an average of 2 m / s (7.2 km / h) per second.

A car that arrives at the red traffic light within Δ t = 3 s < displaystyle Delta t = 3 , mathrm > from "Tempo 50" (v 1 = 50 km h ≈ 14 m s < displaystyle v_ <1> = 50 , mathrm < tfrac > approx 14 , mathrm < tfrac >>) is braked to zero, it is accelerated

Unit of acceleration edit

The unit of measurement for specifying an acceleration is by default the unit Meters per square second (m / s 2), i.e. (m / s) / s. In general, loads on technical devices or the specification of load limits can be given as g-force, i.e. as "force per mass". This is expressed as a multiple of the normal acceleration due to gravity (standard acceleration due to gravity) G = 9.80665 m / s 2 given. In the geosciences, the unit Gal = 0.01 m / s 2 is also used.

Acceleration of Motor Vehicles Edit

In the case of motor vehicles, the positive acceleration that can be achieved is used as an essential parameter for classifying the performance. An average value is usually given in the form “In ... seconds from 0 to 100 km / h” (also 60, 160 or 200 km / h).

For the Tesla Model S (Type: Performance) it is stated that an acceleration from 0 to 100 km / h can be achieved in 2.5 seconds. This corresponds to an average acceleration value of

There are basically two ways of measuring or specifying accelerations. The acceleration of an object can be viewed kinematically in relation to a path (space curve). For this purpose, the current speed is determined, its rate of change is the acceleration. The other option is to use an accelerometer. With the help of a test mass, this determines the inertial force, from which the acceleration is then deduced with the help of Newton's basic equation of mechanics.

Isaac Newton was the first to describe that a force is necessary for an acceleration to occur. His law describes the proportionality of force and acceleration for bodies in an inertial system. An inertial system is a reference system in which force-free bodies move uniformly in a straight line. The acceleration is then the ratio of force F < displaystyle F> to mass m:

If the acceleration is to be calculated in an accelerated reference system, inertial forces must also be taken into account.

Sample calculation for measurement using inertia Edit

In an elevator there is a spring balance on which a mass of one kilogram hangs (m = 1 k g < displaystyle m = 1 , mathrm >). When the elevator is at rest compared to the earth, the balance shows a weight of 9.8 Newtons. The amount of the gravitational acceleration is accordingly

If the spring balance shows a moment later, for example, a force of 14.7 Newtons, the acceleration of the elevator is 4.9 m / s 2 compared to the earth.

General description edit

The acceleration of a body moving along a path (a space curve) can be calculated using Frenet's formulas. This enables an additive decomposition of the acceleration into an acceleration in the direction of movement (tangential acceleration) and an acceleration perpendicular to the direction of movement (Normal acceleration or radial acceleration).

The time derivative of the tangent unit vector can be calculated using the arc length s < displaystyle s>:

The acceleration can thus be broken down into two components:

If the tangential acceleration is zero, the body only changes its direction of movement. The amount of speed is retained. In order to change the magnitude of the speed, a force must act that has a component in the direction of the tangential vector.

Centrifugal acceleration edit

A special case of the above consideration is a circular movement with a constant amount of speed. In this case, the acceleration is directed inwards towards the center of the circle, i.e. always perpendicular to the current direction of movement on the circular path. This special case of a pure one Radial acceleration is called centripetal acceleration. They do not change the amount of the speed, but only its direction, which just results in a circular path. Regarding a co-rotating (and therefore accelerated) Reference system, if an object is accelerated outwards from the center point, then the term centrifugal acceleration is used.

A centrifuge uses this effect to subject things to constant acceleration. The radius of curvature corresponds, since it is a circular movement, the distance r < displaystyle r> of the centrifuged material to the axis of rotation. The acceleration to which the material to be centrifuged is subjected to the path velocity v < displaystyle v> can then also be expressed by the angular velocity ω < displaystyle omega>:

Negative and positive acceleration editing

In the case of a body moving along a line, the tangent unit vector is usually chosen in the direction of movement. If the tangential acceleration is negative, the speed of the body is reduced. In the case of vehicles, one speaks of a deceleration or braking of the vehicle. If the term acceleration is used in this context, it usually means a positive tangential acceleration that increases the speed of the vehicle.

If the initial speed and position are known, the continuous measurement of the acceleration in all three dimensions enables the position to be determined at any point in time. The position can be determined from this simply by double integration over time. In the event that, for example, the GPS device of an aircraft fails, this method enables relatively precise location determination over a medium-long period of time. A navigation system that determines position by measuring acceleration is called an inertial navigation system.

Acceleration field and potential editing

If a force on a particle is proportional to its mass, this is the case with gravitation, for example, it can also be described by an acceleration field. This vector field assigns r → < displaystyle < vec to every location >> in space an acceleration a → (r →) < displaystyle < vec> (< vec >)> to. It can often be expressed as a gradient of a potential Φ (r →) < displaystyle Phi (< vec >)> write. The potential can be clearly understood as a bowl like in the picture on the right. The negative gradient provides a vector that points in the direction of the steepest drop (maximum negative slope). Its direction therefore indicates where a ball that is placed in the bowl would roll. With a potential or acceleration field, the movement of a particle (trajectory) can then be calculated for each initial condition, i.e. initial speed and position.

Even if the force on a particle is not proportional to its mass, a force field and a potential can often be established, for example a Coulomb potential for an electrically charged particle. In this case, however, the acceleration depends on the mass m < displaystyle m> and the charge q < displaystyle q> of the particle:

Constant acceleration edit

With a uniform acceleration, the acceleration field is constant and homogeneous over time, i.e. the acceleration is identical in amount and direction at all points in space, for example equal to the vector g → < displaystyle < vec >> :

With such an approach, the earth's gravitational field can be described locally (not globally). A particle in such a gravitational potential moves on a parabolic path, also called trajectory parabola in the case of a gravitational field. Even with a free fall (without air resistance) all bodies are accelerated equally. On earth, the acceleration towards the center of the earth is approximately 9.81 meters per square second. The earth's gravitational potential, however, is not completely spherically symmetrical, since the shape of the earth deviates from a sphere (earth flattening) and the internal structure of the earth is not completely homogeneous (gravity anomaly). The acceleration due to gravity can therefore differ slightly from region to region. Regardless of the potential, the acceleration due to the rotation of the earth may also have to be taken into account during measurements. An accelerometer used to determine gravitational acceleration is called a gravimeter.

Just as in classical mechanics, accelerations can also be represented in the special theory of relativity (SRT) as the derivation of speed with respect to time. Since the concept of time turns out to be more complex due to the Lorentz transformation and time dilation in the SRT, this also leads to more complex formulations of the acceleration and its connection with the force. In particular, it results that no mass-afflicted body can be accelerated to the speed of light.

The equivalence principle states that there are no gravitational fields in a freely falling frame of reference. It goes back to the considerations of Galileo Galilei and Isaac Newton, who recognized that all bodies, regardless of their mass, are accelerated equally by gravity. An observer in a laboratory cannot tell whether his laboratory is in zero gravity or in free fall. Neither can he determine within his laboratory whether his laboratory is moving uniformly accelerated or whether it is located in an external, homogeneous gravitational field.

With the general theory of relativity, a gravitational field can be expressed using the metrics of space-time, i.e. the measurement rule in a four-dimensional space from spatial and time coordinates. An inertial system has a flat metric. Non-accelerated observers always move on the shortest path (a geodesic) through space-time. In a flat space, i.e. an inertial system, this is a straight world line. Gravitation causes a curvature of space. This means that the metric of the room is no longer flat. This leads to the fact that the movement that follows a geodesic in the four-dimensional space-time is usually seen by the outside observer in the three-dimensional visual space accelerated movement is perceived along a curved curve.

Magnitude of typical accelerations from everyday life: [1]

  • The ICE reaches an acceleration of around 0.5 m / s 2, a modern S-Bahn railcar even 1.0 m / s 2.
  • During the first steps of a sprint, accelerations of around 4 m / s 2 act on the athlete.
  • The acceleration due to gravity is 9.81 m / s 2.
  • In the shot put, the ball is accelerated at around 10 m / s 2 in the push-off phase.
  • In a washing machine, more than 300 work in the spin cycle G (≈ 3,000 m / s 2) on the contents of the drum.
  • A tennis ball can experience accelerations of up to 10,000 m / s 2.
  • In the case of nettle cells, the sting is up to 5,410,000 G (≈ 53 million m / s 2) accelerated.

At drag racing tracks, among other things. measured the time for the first 60 feet. While very fast road vehicles like the Tesla Model S P90D need around 2.4 seconds for this, a Top Fuel Dragster typically passes the mark in less than 0.85 seconds. The finish line at 1000 feet, a good 300 meters, is passed in under 3.7 seconds at over 530 km / h.

The shortest time from zero to 100 km / h in Formula Student was achieved in June 2016 by the “grimsel” electric racing car built by students from the ETH Zurich and the Lucerne University of Applied Sciences, which reached 100 km / h in 1.513 seconds on the Dübendorf military airfield near Zurich and reached within less than 30 meters, setting a new world record for electric vehicles. [2]

1 answer

According to the law of conservation of energy, neither energy is lost, nor is additional energy added out of nowhere. If the friction is neglected, the ball will again reach the same height as at the start (same potential energy).

In between, the potential energy is converted into kinetic energy (movement) by the acceleration of gravity and then back into potential energy. Assuming that the center of the sphere corresponds to the center of gravity and starts with height h2, the sphere reaches height h2 again.

The difference between energy and work: energy is a state, work is a process. Example. The energy that the sphere has is not in it from the start. Somebody had to lift the ball to its starting position. Lifting is a process, it is the work that had to be done in order for the ball to get its energy. If you had invested more work and placed the ball in an even higher position, it would also have a higher potential energy (of the position).

Calculating acceleration in physics: formula + examples

When calculating the acceleration using the formula in physics, it is best to always follow the steps explained here. For acceleration we need ours derived formula of uniform motion just add a few factors.

There are different formulas, depending on how many factors play a role, for example whether the acceleration starts from a standstill.

We are dealing here with the uniform standard cases of grades 5-7:

where a is the acceleration, v is the speed and t is the time.

It is very important that you can safely convert the units to solve such tasks. Let's look at an example:

A sports car (1) accelerates from 0 to 180 km / h in 40 seconds, an off-road vehicle (2) can reach a speed of 150 km / h in 70 seconds. Which of the two cars accelerates faster?

  • Because the acceleration according to the SI units has been set to the unit m / s ^ 2 you should convert all given quantities into m and s in the first step:
  • You only have to do this Transformation factorknow:
  • 180 km / h = 180 km * 1000 (in meters) / 1h * 3600 (in seconds) = 180 * 1000/3600 = 50 m / s
  • 150 km / h = 150 km * 1000/3600 = 41.66 m / s

Now we simply insert our transformed quantities into the formula of the acceleration:

So we see that the sports car accelerates more than twice as fast as the off-road vehicle.

Example 2:

A rocket accelerates from a standing start for 0.34 minutes evenly with a = 60 m / s ^ 2. What is the final speed in km / h after the time?

Description Newton's second law - F = m · a

Why are racing bikes so much lighter than a normal everyday bike or a mountain bike?

With the help of an experimental set-up, I will show you how the quantities force, acceleration and mass are connected and what significance Newton's basic law has.

Transcript Newton's second law - F = m · a

Basic law of dynamics: Newton 2

Hello. Did you know that racing bikes on the Tour de France only weigh half as much as mountain bikes? There is even a competition regulation stating that the mass of the racing bikes must not be less than 6.8 kg. Can you imagine what impact the crowd could have on the competition?

The search for the answer leads us to the basic law of dynamics, also known as Newton's second axiom.

For our investigation, we first set up a hypothesis on the questions. Then I'll show you an experimental setup with which we can determine the relevant parameters well. The evaluation of the experiment brings us to the basic law of dynamics.

As you already know, one of the effects of a force is the change in the state of motion of a body. This means that either the amount of speed of the body is increased or decreased, or the direction of movement is changed. You have also already learned that the change in speed is called acceleration and that there is a time-distance law for this.

When cycling, the muscle power of the legs acts on the pedals, which drives the rear wheel and, thanks to the friction between the road and the tire, pushes the bike in the direction of movement and thus accelerates it. The acceleration and the force act in the same direction. And we now know from racing bikes that the mass of the bike should be particularly small.

So for our experiment we hypothesize that the acceleration of a body is greater, the greater the force acting and the smaller the body's mass.

Since measuring acceleration and force on a bicycle is a bit difficult, I'll show you how to examine them in more detail in an experimental setup. An air cushion track is best because there is almost no friction here. If you don't have the air track, you can also use a trolley on casters. There are several pieces of weight on the wagon, so we have our bike plus driver. The force on the car is realized by an attached mass piece. The weight of the mass piece acts as a tensile force on the carriage via the pulley. Complete. With a ruler and a stopwatch we can now measure the change in distance and time.

If we change the time-distance law of uniformly accelerated motion according to a, then we get a is equal to two s by t squared. With this equation we can calculate the mean acceleration of the car.

Since we want to investigate how the acceleration changes with different forces, we now have to take weights from the cart and hang them on our string. The accelerating force increases while the total mass remains constant. We carry out a new measurement for each additional weight.

We enter the resulting pairs of values ​​in a diagram. The force in Newtons is plotted on the right axis and the acceleration in meters per square second on the high value axis.

If we now enter the value pairs, we can clearly see the connection. We can put a straight line through the origin through the points, which means that there is direct proportionality here. The acceleration is therefore directly proportional to the force. This proportionality even applies to the vector quantities since force and acceleration point in the same direction, but in this diagram we can only enter the amounts. This would confirm the first part of our hypothesis.

Next, let's examine how the acceleration changes when the mass of the bike changes. To do this, we use the same experimental set-up as before, but this time we start with a piece of mass on the cart and one on the string. Then we gradually put the extra weights on the cart. The accelerating force now remains constant, while the mass now changes.

We carry out a measurement for each additional piece of mass and put the value pairs in a diagram. This time, of course, the total mass in kg is plotted on the right axis.

When drawing in the points, it quickly becomes clear that this time there is no direct proportionality. This time the curve is a hyperbola. The larger the mass, the smaller the acceleration. This relationship is called indirect proportionality. The acceleration is inversely proportional to the mass. So our hypothesis has proven to be absolutely correct.

Isaac Newton also came to these two insights and one of his numerous achievements was to formulate a common basic law from them. The acceleration is equal to the force through the mass. Or rearranged: A force F causes a body of mass m to experience acceleration a. This law also applies in vector notation, since force and acceleration point in the same direction.

This formula is also great for determining the unit of force. The unit of mass is kilograms, that of acceleration is meters per square second. And so a newton is made up of kilograms times meters divided by a square second.

The mass m is to be understood here as inert mass. One can imagine that the inertia of the mass acts as a resistance to acceleration and hinders it. There is also the heavy mass, which plays a role in gravitation and causes masses to attract.

So you see: force, mass and acceleration are closely related. In order to achieve the greatest possible acceleration, racing bikes are built to be particularly light. The drivers are usually quite slim, but have strong legs so that they can pedal powerfully. We find Newton's basic law of dynamics everywhere in our everyday life. Wherever bodies are moved. Until next time!

What happens if a force is not acting parallel to the direction of movement?

If the force does not act parallel to the direction of movement, the direction of travel will change.
Background: Accelerations always occur when forces act on a body. If such a force acts parallel to the current direction of movement, there is no change in direction and only an increase or decrease in speed (e.g. when accelerating or braking a vehicle on a straight road).

Video: Προσδιορισμός της επιτάχυνσης της βαρύτητας (August 2022).