Acceleration of acceleration?

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In summary, the conversation discusses the concepts of velocity, acceleration, and acceleration of acceleration. The idea of calculating the exact acceleration in any given moment is proposed, and it is compared to the traditional method of using average acceleration. The concept of "jerk" is also mentioned, which is the third derivative of position with respect to time. Different approaches to solving equations involving acceleration are also mentioned. The conversation ends with a practical example of experiencing acceleration and the importance of being aware of "jerk" in everyday situations.
  • #1
chocolatesheep
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Hi everyone.

I was wondering about this.
If the position of an object changes in time, the object has velocity.
If the velocity of an object changes in time, the object is accelerating (decelerating)
If the acceleration of an object changes in time, could we hypothetically have acceleration of acceleration.

I have this scenario in my head:
An asteroid is at rest, far away from earth. The Earth starts dragging the asteroid towards it. The asteroid will accelerate towards Earth ever so slightly (since gravity depends on distance)
The closer it gets to earth, not only will it go faster, but it will also accelerate faster.
So we will have the change of acceleration in time:
[itex]a\prime = \lim_{\Delta t\to\ 0}\frac{\Delta a}{\Delta t}[/itex]

So acceleration of acceleration would be in [itex]\frac{m}{s^3}[/itex] or rather [itex]\frac{\frac{m}{s^2}}{s}[/itex] if it appeals more.

Would this be useful? We could calculate the exact acceleration in any given moment as opposed to having the average acceleration.
But that's beside the point since we could always calculate the exact acceleration if we know how far it is from a planet.

But if we needed to know the exact speed of the asteroid after let's say 20 hours, we would get an incorrect answer if we treated the acceleration as if it were constant.

So [itex]a_1 = a_0 \pm a\prime t[/itex]
so if [itex]a_0 = 0[/itex] then [itex]a_1 = a\prime t[/itex]

and if [itex]v_1 = v_0 \pm at[/itex]
and if [itex]v_0 = 0[/itex] then [itex]v_1 = at[/itex]

=> [itex]v_1 = a\prime t^2[/itex] if the object is starting to move from rest

Does any of this make sense?
Waiting for someone to point out a flaw in this.
 
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  • #2
I didn't read through but a' is called the "jerk".
 
  • #3
After "jerk", the next three derivitives are affectionately known as "snap", "crackle" and "pop".

Ordinarily one does not attack these kinds of problems by looking at higher and higher order derivitives though. Instead one writes down a differential equation that relates, for instance, acceleration to velocity and position.

One can use differential calculus to attempt to solve such an equation, reducing it to a form that expresses position as a function of time.

Failing that, there are computational methods (such as Runge Kutta) that generate approximate solutions by running a kind of simulation and advancing stepwise. Such approaches often treat the second derivitive (acceleration) as a variable and work in part by estimating its average value over the duration of each small step.

This sounds similar to what you are talking about.
 
  • #4
You can do a simple experiment to 'feel' this when braking your automobile.

As you get it almost stopped, lighten pressure on the brake so you don't slam back into seat when it stops, instead making a gradual approach to rest.

Acceleration with a negative sign is the force pushing you forward. Your passengers will appreciate your awareness of and attention to "jerk".
 
  • #5


I can say that your understanding of acceleration and its relationship to velocity and position is correct. It is also possible to have a changing acceleration, which can be represented by the concept of "jerk" in physics. However, the term "acceleration of acceleration" is not commonly used in the scientific community.

In your scenario with the asteroid, what you are describing is the changing acceleration due to the changing distance between the asteroid and Earth. This is a result of the gravitational force between the two objects and can be calculated using the laws of gravity. In this case, we would not use the term "acceleration of acceleration" but rather just describe it as a changing acceleration due to the changing distance.

In general, we do not need to calculate the exact acceleration at any given moment because we can use calculus to calculate the average acceleration over a specific time interval. This is often sufficient for most scientific and engineering purposes. However, in some cases where extreme precision is needed, considering the changing acceleration may be necessary.

Overall, while the concept of acceleration of acceleration is not commonly used, it is important to understand the relationship between acceleration, velocity, and position and how they can all change over time.
 

What is acceleration of acceleration?

Acceleration of acceleration, also known as jerk, is the change in acceleration over time. It measures the rate at which acceleration is changing.

How is acceleration of acceleration related to acceleration?

Acceleration of acceleration is the second derivative of displacement. In other words, it is the rate of change of acceleration, just as acceleration is the rate of change of velocity.

Why is acceleration of acceleration important?

Acceleration of acceleration is important because it can affect the stability and comfort of moving objects. High levels of acceleration of acceleration can cause discomfort and even injury to humans, and can also impact the performance and efficiency of machines and vehicles.

How is acceleration of acceleration measured?

Acceleration of acceleration can be measured using a variety of methods, such as accelerometers, gyroscopes, and other sensors. These devices can track the changes in acceleration over time and calculate the acceleration of acceleration.

What are some real-world examples of acceleration of acceleration?

Some examples of acceleration of acceleration in the real world include roller coasters, where the high levels of jerk can cause discomfort for riders, and rocket launches, where the sudden changes in acceleration can put stress on the spacecraft and its occupants.

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