All Questions
Tagged with differentiation velocity
105 questions
57
votes
7
answers
10k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,883
34
votes
7
answers
5k
views
The usage of chain rule in physics
I often see in physics that, we say that we can multiply infinitesimals to use chain rule. For example,
$$ \frac{dv}{dt} = \frac{dv}{dx} \cdot v(t)$$
But, what bothers me about this is that it raises ...
- 8,040
26
votes
4
answers
6k
views
With what velocity are we moving along the time dimension?
Does the question make sense? Velocity along time axis means $v_t=\mathrm dt/\mathrm dt$? If it doesn't, please explain where the flaw is. Taking time as measure like length? Or do we need to ...
- 659
17
votes
7
answers
6k
views
What's the difference between average velocity and instantaneous velocity?
Suppose the distance $x$ varies with time as:
$$x = 490t^2.$$
We have to calculate the velocity at $t = 10\ \mathrm s$.
My question is that why can't we just put $t = 10$ in the equation $$x = 490t^2$...
15
votes
2
answers
4k
views
Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?
Why is the following equation true?
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$$
where $\mathbf{v}_i$ is velocity, $\mathbf{r}_i$ is the ...
- 1,483
11
votes
4
answers
3k
views
When the direction of a movement changes, is the object at rest at some time?
The question I asked was disputed amongst XVIIe century physicists (at least before the invention of calculus).
Reference: Spinoza, Principles of Descartes' philosophy ( Part II: Descartes' Physics, ...
user229097
10
votes
7
answers
1k
views
What is the instant velocity? [duplicate]
The velocity is the variation rate of the position correct? So does it make sense to talk about velocity without time?
- 117
9
votes
4
answers
4k
views
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant but acceleration is constant?
Is it ever possible that the object is moving with a velocity such that its rate of change of speed is not constant, but rate of change of velocity is constant?
Like speed is only the magnitude, so ...
9
votes
4
answers
2k
views
Can I find the acceleration or velocity when my displacement-time graph is discontinuous?
Today, I encountered the problem where I was asked to find the velocity and acceleration from displacement-time graph but the displacement-time graph was discontinuous. So I am unable to find the ...
- 123
6
votes
6
answers
1k
views
Question about derivation of kinematics equations
Apologies if this has been asked before, but I browsed the sub and couldn't find something specific.
I understand the derivation for one of the equations as follows:
\begin{gather}
\frac{dv}{dt} = a ...
- 69
5
votes
2
answers
2k
views
How does instantaneous velocity or acceleration have any other numerical value than 0? [duplicate]
Instantaneous velocity is defined as the limit of average velocity as the time interval ∆t becomes infinitesimally small. Average velocity is defined as the change in position divided by the time ...
- 163
5
votes
4
answers
5k
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How can there be really any instantaneous velocity?
I have read about Zeno's arrow paradox that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we ...
user36790
4
votes
6
answers
856
views
How to understand instantaneous velocity concept [duplicate]
When I started learning instantaneous velocity it didn't make sense to me. I don't understand in real life why we can't measure instantaneous velocity and therefore why we use this concept.
Or is this ...
- 311
4
votes
4
answers
413
views
Suppose there is a vector $\vec v$ which is a function of time, then will $\dfrac{d}{dt}|\vec v|$ be a vector quantity or a scalar quantity?
Suppose there is a vector $\vec v$ which is a function of time, then will $\dfrac{d}{dt}|\vec v|$ be a vector quantity or a scalar quantity?
I think it should be scalar because, let's assume $\vec v=...
- 223
4
votes
2
answers
594
views
Hamilton's Formulation and Independent Coordinates
In Lagrange's formulation we know that $q,\dot {q}$ are independent of each other i.e,
$$\frac { \partial q }{ \partial \dot { q } } =0.$$
My question is, is this true for $p$, $q$ in Hamilton's ...
- 588
3
votes
9
answers
4k
views
Can velocity be an undefined quantity?
We have the image below displaying the uniform velocity by time-distance graph. At every point velocity is constant but what if distance and time both become zero as at origin in the graph is? The ...
user66452
3
votes
3
answers
296
views
If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined? [closed]
If instant velocity at any given time $t_0$ is defined as the derivative of $x(t)$ at $t_0$, what if the derivative does not exist? How are we supposed to deal with $x(t)=|t|$ at $t=0$, or for more ...
- 39
3
votes
2
answers
267
views
What does $\dot x$ mean as an operator in quantum mechanics?
I've been looking at a paper titled "Feynman's proof of the Maxwell Equations" by Freeman Dyson (American Journal of Physics 58, 209 (1990); https://doi.org/10.1119/1.16188) and I'm confused ...
- 41
3
votes
1
answer
4k
views
What is the meaning of word 'rate' in physics?
Often, I have seen in physics the rate of change of velocity or something like that in kinematics. And in question based on speed, time and distance. I would like to know the meaning of the word rate ...
3
votes
2
answers
233
views
Generalization of straight line motion under constant acceleration
My question is that, we all know the three equations of straight line motion under constant acceleration,
\begin{align}
x & =x_{\rm o}+v_{\rm o}\,t+\tfrac12 \mathrm a\,t^2
\tag{1d-a}\label{1d-a}\\
...
3
votes
3
answers
2k
views
Physical Meaning of Divergence of Convective Velocity Term
When taking the divergence of the convective velocity term, I get the following:
\begin{align}
\nabla\cdot\left[\mathbf u\cdot\nabla\mathbf u\right]&=\frac{\partial}{\partial x_i}\left[u_j\frac{\...
- 379
2
votes
5
answers
346
views
Significance of $\frac{dv}{dx}=0$
Suppose an object is moving with varying acceleration in time.
What does it mean when it hits a point where $\frac{dv}{dx}=0$?
Does it mean the object has hit maximum velocity?
Assume the object ...
- 77
2
votes
4
answers
318
views
Why is $\vec{v}\cdot d \vec{v} = v dv$? [closed]
Can someone help me understand why is this true:
$$\vec{v} \cdot d \vec{v} = v \cdot dv$$
where $v$ is speed? I found somewhere that $\vec{v} \cdot d \vec{v}=|\vec{v}||d \vec{v}| \cos \phi$. And I ...
- 37
2
votes
4
answers
733
views
Can a particle have no instantaneous velocity at all points of the path taken but a finite average velocity?
I have a question on kinematics.
Say the path traced by a particle is given by a Koch curve or Koch snowflake.
Now consider the particle starts from some arbitrary point $A$ on the curve and ...
- 4,984
2
votes
3
answers
193
views
Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?
My teacher has proved the following:
$$ \mathit v^2 = \mathbf v·\mathbf v = \frac{d\mathbf r}{dt}·\frac{d\mathbf r}{dt} = \left(\frac {ds}{dt}\right)^2 \Rightarrow \mathit v = \frac{ds}{dt} $$
Because ...
- 23
2
votes
4
answers
668
views
Interpretation of Velocity as a time derivative of position
Going by the Wikipedia explanation, a derivative measures the 'sensitivity' of a function to tiny nudges in its input.
How well does this fit with the velocity being the derivative of position? I can'...
- 129
2
votes
4
answers
20k
views
How to find tangential/radial/angular velocity for motion in any curve? [closed]
Is the radial velocity responsible only for changing distance between objects and the component perpendicular to it only for change in direction? If so why?
Please try to give a different explanation ...
- 188
2
votes
3
answers
198
views
What is the definition of velocity?
We know that displacement is change in an object's position (here position means 'position vector'). Then velocity will be change in position of the object with respect to time, simply displacement/...
2
votes
1
answer
292
views
Is the relation "slope=velocity" mathematically valid?
$\text{Slope= tan(angle with respect to positive X-axis)= scalar output}$
$\text{velocity= a vector }$
Source: Hugh D Young_ Roger A Freedman - University Physics with Modern Physics In SI Units (...
- 439
2
votes
1
answer
267
views
Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?
Consider a particle that moves around the coordinate grid. After $t$ seconds, it has the position
$$
S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, .
$$
The particle traces a quarter arc of ...
- 131
2
votes
1
answer
203
views
Time derivative with respect to an observer moving with velocity $\mathbf{v}$
I am taking a class in fluid mechanics right now and my book has this statement with no explanation:
What is the time derivate seen by an observer moving with a velocity $\mathbf{v}$ of a scalar ...
- 707
2
votes
3
answers
179
views
Difference between $|d{\bf r}|$ and $d|{\bf r}|$
What is the difference between $|d{\bf r}|$ and $d|{\bf r}|$ and why are both of them not always equal to each other?
My question might seem stupid to some and will probably get downvoted but I have ...
- 733
2
votes
2
answers
622
views
Instantaneous velocity applications
I refered these two questions
Instantaneous velocity
How to interpret instantaneous velocity using limit?
and I understood how instantaneous velocity is defined. But why do we define it?
Velocity ...
- 169
2
votes
1
answer
435
views
When exactly does velocity increase or decrease on an acceleration time graph? [closed]
How does the acceleration time graph show if and object is speeding up or slowing down?
Is it possible to find the answer without any deep calculations? If yes then how?
Like how can I find the ...
- 137
1
vote
2
answers
319
views
What is the time derivative of the linear velocity vector $\vec{v}\,(t)$?
If $\vec{v}\,(t)$ denotes linear velocity, we can then write $\vec{v}\,(t)$ as $|v(t)|\hat{v}$. My question is what is $\displaystyle\frac{d\vec{v}\,(t)}{dt}?$
The answer I have seen to this question ...
- 39
1
vote
2
answers
220
views
Average velocity and instantaneous velocity
In some books of Physics in Italian language, they write that the instantaneous velocity $v$, is:
$$v=\frac{dr}{dt}=\lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t}$$
where $v_{\text{avg}}={\Delta r}/...
- 2,575
1
vote
4
answers
6k
views
When we take time derivative of a function of time, then is the result another function of time, again?
(I'll try to explain my question by one known example), for example where the velocity is a function of time v(t) then its time derivative (which is acceleration: $a=\frac {dv}{dt}$) is another ...
- 21
1
vote
3
answers
95
views
What is the rate of change of time wrt velocity of an object?
disclaimer, I'm just an average highschooler so please be a little friendly with the mathematics of your answers but I wondered what would be $dt/dv$?
- 33
1
vote
1
answer
172
views
$dT/dx=0$ always true?
In a Classical Mechanics book I found the assumption that for an arbitrary particle with constant mass in the Real line $dT/dx=0$, with T the Kinetic Energy i.e. $T=(m·\dot x^2)/2$
My hypothesis is ...
- 145
1
vote
5
answers
385
views
A Problem with velocity vector
I am having a conceptual problem. I understand why the definition of the velocity of a body moving in one dimension is the derivate of its position coordinate. But I don't get why the velocity vector ...
- 13
1
vote
2
answers
557
views
In the equation: $a = dv/dt$ , is $dt$ the time taken to achieve that instantaneous acceleration?
If you solve for $dt$ from $a = \frac{dv}{dt}$ , is it the time taken to to achieved that instantaneous acceleration?
$a$ : acceleration
$v$ : velocity
$t$ : time
- 87
1
vote
2
answers
142
views
Average velocity showing different results
I was solving a question, in which, a particle has travelled a distance $s$, with initial velocity $0$ and constant acceleration.
So the equation of motion becomes,
$$ v = a t \tag{1} $$
and
$$ v = \...
- 56
1
vote
1
answer
348
views
Time derivative of $\rm{atan2}$ when $x=0$
I want to take the time derivative of the $\rm{atan2}$ function to calculate an azimuth rate in spherical coordinates, given position and velocity in Cartesian $xyz$ coordinates.
$$\rm{atan2}(y, x) =
\...
- 206
1
vote
5
answers
7k
views
Direction of velocity vector in 3D space
According to a well-known textbook (Halliday & Resnick), the direction of a velocity vector, $\vec v$, at any instant is the direction of the tangent to a particle's path at that instant, as is ...
- 113
1
vote
3
answers
647
views
Derivative as a fraction in deriving the Lorentz transformation for velocity
Consider a frame $S$ and $S'$ which is coincides at $t=0$ and then $S'$ starts moving with velocity $v$ in $+x$ direction.
By Lorentz transformation equation,
\begin{align}
x'&=\gamma(x-vt) \\
...
- 446
1
vote
4
answers
58
views
Can we calculate centripetal acceleration by using this method $\frac{\mathbf v_2-\mathbf v_1}{T}$?
If we know the angle between two velocity vectors $\mathbf v_1$ and $\mathbf v_2$, and if we know the time $T$ it takes for the velocity to change from $\mathbf v_1$ to $\mathbf v_2$,then is it ...
- 315
1
vote
3
answers
2k
views
How does instantaneous speed work for circular motion?
Why do we use the formula $\lim_{\delta t→0} \delta s/\delta t$ to get the instantaneous speed? Since speed is distance divided by time, what does the limit have to do with this? I have a very limited ...
- 341
1
vote
2
answers
167
views
Velocity and acceleration in special relativity
I would like to compute what the constant acceleration trajectories are in the Minkowski spacetime $(t, x)$ with $d\tau^2 = dt^2 - dx^2$. So given some trajectory $x(t)$ I know the velocity vector is ...
- 592
1
vote
2
answers
3k
views
Velocity time graph analysis: what does a concave downward $v$-$t$ curve mean?
This is a screenshot from the lecture about the analysis of various velocity-time graphs I was watching.
I understand that
the concavity of velocity-time graph will tell about the
increasing or ...
- 646
1
vote
0
answers
93
views
Does car move when instantaneous velocity is zero? [duplicate]
In 3blue1brown: derivative paradox.
supposed car moving with:
$S(t) = t^3$
And velocity is:
$V(t) = 3t^2$
He asked when t = 0 velocity is 0 m/s , does that car move at that time ?
And here his ...
- 311