What Does The Laplacian Of A Fluid Velocity Potential Mean
I was studying something related to fluid mechanics and then I found that $\nabla^2 \Phi = 0$ where $\Phi$ is the fluid velocity potential . So I was wondering what does it mean that the laplacian of the fluid velocity potential is equal to zero and also in general.
I know the gradient of a scalar function tells us about the direction in which the function increases and the divergence of a field about how it spreads out in space. Then I could guess that the Laplacian is equal to zero because the fluid could be uniform and thus the velocity field doesn’t spread out.
Is this a correct assumption?
One of the consequences of $\nabla\cdot V = 0$ is the incompressibility of the fluid. It is perfectly possible for air, for example, that some portions of space are temporary being depleted $\nabla\cdot V > 0$ or filled $\nabla\cdot V < 0$.
edit :if V can be expressed as a gradient of some scalar, its rotational is:
$\nabla$ X $V$ = $\nabla$ X $ =$
$\frac – \frac = \frac – \frac = \frac – \frac = 0$
The same for the other components.
Resuming: $\nabla\cdot V = 0$ means incompressibility, but the additional condition of $\nabla^2 \phi = 0\ $, \ $means the fluid has rotational = 0 everywhere.
Average Vs Instantaneous Speed
The examples so far calculate average speed: how far something travels over a period of time.
But speed can change as time goes by. A car can go faster and slower, maybe even stop at lights.
So there is also instantaneous speed: the speed at an instant in time. We can try to measure it by using a very short span of time .
Acceleration And Newtons Second Law
Acceleration, rather than velocity, forms a key part of Newtons second law of motion. The equation is F = ma, where F stands for force, m is mass, and a is the acceleration. Because of the link between velocity and acceleration, you can also write this as force = mass × the rate of change of velocity. However, acceleration is the key characteristic here, not velocity.
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Difference Between Speed And Velocity
The detailed comparison in the tabular format is given below.
|Velocity VS Speed|
|Speed is the quantitative measure of how quickly something is moving.||Velocity defines the direction of the movement of the body or the object.|
|Speed is primarily a scalar quantity||Velocity is essentially a vector quantity|
|It is the rate of change of distance||It is the rate of change of displacement|
|Speed of an object moving can never be negative||The velocity of a moving object can be zero.|
|Speed is a prime indicator of the rapidity of the object.||Velocity is the prime indicator of the position as well as the rapidity of the object.|
|It can be defined as the distance covered by an object in unit time.||Velocity can be defined as the displacement of the object in unit time.|
Lets summarize what we have learned:
- This is a function of displacement.
- We require both magnitude and direction to define velocity.
- Instantaneous velocity is the velocity at any given moment of time, whereas average velocity is the total displacement divided by total time.
- A changing velocity indicates acceleration.
Constant Acceleration Vs Constant Velocity
Traveling with a constant velocity means youre going at the same speed in the same direction continuously. If you have a constant velocity, this means you have zero acceleration. You can imagine this as driving down a straight road but keeping your speedometer on the same value.
A constant acceleration is quite different. If you travel with a constant acceleration, your velocity is always changing, but its changing by a consistent amount each second. The acceleration due to gravity on the Earth has the constant value 9.8 m/s2, so you can imagine this like dropping something from a skyscraper. The velocity starts low, but increases by 9.8 m/s for every second it is falling under gravity.
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How Is The Vector Constant
When you use a scalar as a constant. In that case, only the value of the scalar quantity needs to be kept unchanged over time.
For example, your room temperature is 30 °C and you notice that as time increases, your room temperature stays at 30 °C. Then you can say that your room temperature is constant.
But, vector quantity has both value and direction. Thus, if the vector is to be constant, the direction must remain unchanged with the value. That is, the value of the vector will not only be constant over time but also the direction of the vector will remain unchanged.
Suppose a particle is moving along the x-axis with a velocity of 30 m/s. And you notice that after the specified time the particle continues to move along the x-axis at the same speed. In this case, both the value and the direction of velocity are constant even after the particle has crossed the specified path. Then we can say that the particle is moving with constant velocity.
Suppose the particle of the above problem started moving along the y-axis with the same velocity after a certain time without going along the x-axis. In this case, the direction of the particles velocity will change after crossing the specified path. Then the velocity of the particle will not be constant after the specified time.
What Does A Straight Diagonal Line Mean In Physics
4.4/5straight diagonal linelineline
Just so, what does a diagonal line going back to zero represent on a distance time graph?
And if its motion is all in the same direction, it also has constant velocity.
Subsequently, question is, what is the slope of St graph? The gradient of a distance-time graph represents the speed of an object. The velocity of an object is its speed in a particular direction. The slope on a velocity-time graph represents the acceleration of an object.
Moreover, what does the slope represent in physics?
The slope of a velocity graph represents the acceleration of the object. So, the value of the slope at a particular time represents the acceleration of the object at that instant.
What does a diagonal line on a distance time graph mean?
Distance–time graphs. If an object moves along a straight line, the distance travelled can be represented by a distance–time graph. On a distance–time graph, the gradient of the line is equal to the speed of the object. The greater the gradient the faster the object is moving.
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The Function Of Velocity
The difference between an objects starting point and its ending location is its amount of displacement. Two possible values of velocity are an objects average velocity and its instantaneous velocity . You can measure velocity in the following units:
- m/s – meters per second
- kph – kilometers per hour
- mph – miles per hour
If an objects velocity is lower than its speed, you can tell that it did not travel in one direction. You need to measure an objects velocity before you can record any increases in its speed also known as acceleration.
Related Questions And Answers
What Is The Difference Between Velocity And Acceleration
Velocity and acceleration both describe motion, but there is an important difference between the two. If youre studying physics at high school or college level, understanding the differences between them is essential. Understanding what velocity means leads to an understanding of what acceleration means because while velocity is the rate of change of position, acceleration is the rate of change of velocity. If youre traveling at a constant pace, you have velocity but no acceleration, but if youre traveling and your pace is changing, you have velocity and acceleration.
Velocity is the rate of change of position with respect to time, whereas acceleration is the rate of change of velocity. Both are vector quantities , but the units of velocity are meters per second while the units of acceleration are meters per second squared.
Constant Velocity Vs Acceleration
To have a constant velocity, an object must have a constant speed in a constant direction. Constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed.
For example, a car moving at a constant 20 kilometres per hour in a circular path has a constant speed, but does not have a constant velocity because its direction changes. Hence, the car is considered to be undergoing an acceleration.
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Measuring Our Lives With Physics
Knowing how fast youre going or are capable of going is helpful for drivers, cyclists, runners, and anyone who can read a speedometer. But speed is just the starting point for many valuable concepts in physics. Check out additional resources on velocity, including more examples of scalar and vector quantities and how velocity differs from acceleration.
Velocity In Real Life
While it’s true that you’ll probably not be drawing graphs to explain velocity in real life, velocity does actually have real-life applications. Maybe you’ll take a road trip and need to know how to figure out how much time it will take you to get somewhere. Likewise, people design airports using formulas like these to help them figure out how long the runway needs to be so the airplane has enough space to take off and land. No doubt about it, velocity is a practical part of everyday physics.
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Velocity And Acceleration Equations
Finding an objects average velocity means that you know its displacement, or change in position , the change in time during its journey , and the direction traveled. The equation for average velocity looks like this:
v = s/t
It gets a bit more complicated when calculating an objects average acceleration . For that, youll need to know its change in velocity and the change in time . It looks like this:
a = v/t
Basically, you are figuring out the rate of change in meters per second squared . This solution will tell you the proper acceleration for an object over a set period of time.
The Main Difference Between Velocity And Acceleration
Velocity and acceleration both use speed as a starting point in their measurements. Speed, which is the measurement of distance traveled over a period of time, is a scalar quantity. Both velocity and acceleration are vector quantities, meaning that they use both magnitudes and a specified direction.
Here are the basic definitions of velocity and acceleration:
- velocity – the rate of displacement of a moving object over time
- acceleration – the rate of velocity change over time
As you can see, velocity requires speed for its measurement, and acceleration requires velocity for its measurement. In order to properly measure any of these quantities, youll need to know how both velocity and acceleration work.
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Instantaneous Velocity And Instantaneous Speed
When we talk about the standard equation, almost always, we are talking about average speed and average velocity. However, sometimes it is necessary to know the velocity or speed at an instant.
The velocity of an object at a specific point in time is known as instantaneous velocity. The general equation for instantaneous velocity is:
The speed of an object at a specific point in time is known as instantaneous speed. The general equation for instantaneous speed is:
x = Displacements = Distance
The most common example of the measurement of instantaneous speed is a speedometer. Speedometers show the speed of an object in a particular instant, not its average value.
Can The Speed And Velocity Of An Object Be Different
There is always a comparison drawn between speed and velocity regarding the motion of an object. These two entities are constantly pitted against each other to see which is superior.
But there is no point comparing vector and scalar quantities because they are distinct in their ways.
To give a rough idea, consider an object traveling in a perfect circle at 20 kmph. Since this is a perfect circle and we have defined a constant rate of motion for it, speed and velocity should be the same, right?
Wrong! In this case, the speed of the object is constant, but the velocity is always changing.
This is because we consider direction when defining velocity. So when an object travels in a circle, we find that its direction is always in a state of change.
If we find out the direction of that object at any given point, it will be tangential to the outline edge of that circle.
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What Is Constant Velocity
When you read mechanics, you will hear the words uniform velocity or constant velocity.
These two words mean the same thing. That is, constant velocity is called uniform velocity. There are many concepts of mechanics hidden in this Uniform Velocity word. So, in this tutorial, we will discuss in depth about constant-velocity through graphical representation.
Before discussing in detail you need to understand when a vector quantity is constant because velocity is a vector quantity.
What Does Velocity Mean In Motions Graphs
Velocity is the change in position occurring over a change in time. Change in position is known as displacement, and is represented by #Deltad# , and change in time is represented by #Deltat# , and velocity is represented by #/# .
In position vs. time graphs, time is the independent variable and is on the x-axis, and position is the dependent variable and is on the y-axis. The velocity is the slope of the line, and is the change in position/change in time, as determined by #/# #/# .
The following position vs. time graph shows the different possibilities when the velocity is constant. Constant velocity is represented by a straight line on a position vs. time graph.
On the graph, Line A represents constant negative velocity. Lines B and D represent constant positive velocity. The steeper slope of Line B indicates a faster velocity than D. Line C indicates a constant velocity of zero, meaning the object is at rest.
The position vs. time graph below indicates that the motion of an object is not constant. Suppose it is a car. For the first 10s, it travels at a constant positive velocity. For the next 5s, its velocity is zero, meaning it has stopped. For the next 25s it travels at a constant negative velocity, and for the last 15s, it travels at a constant positive velocity and returns to its initial position.
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Initial And Final Velocity
Initial velocity describes how fast an object travels when gravity first applies force on the object. On the other hand, the final velocity is a vector quantity that measures the speed and direction of a moving body after it has reached its maximum acceleration.
How to find the final velocity?
Finding the final velocity is simple with a few calculations and basic conceptual knowledge.
The Average Velocity Formula And Velocity Units
The average velocity formula describes the relationship between the length of your route and the time it takes to travel. For example, if you drive a car for a distance of 70 miles in one hour, your average velocity equals 70 mph. In the previous section, we have introduced the basic velocity equation, but as you probably have already realized, there are more equations in the velocity calculator. Let’s list and organize them below:
final velocity = initial velocity + acceleration * time
average velocity = velocityâ * timeâ + velocityâ * timeâ + …
You should use the average velocity formula if you can divide your route into few segments. For example, you drive a car with a speed of 25 mph for 1 h in the city and then reach 70 mph for 3 h on the highway. What is your average velocity? With the velocity calculator, you can find that it will be about 59 mph.
From the above equations, you can also imagine what are velocity units. British imperial units are feet per second ft/s and miles per hour mph. In the metric SI system the units are meters per second m/s and kilometers per hour km/h. Remember you can always easily switch between all of them in our tool!
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