All Questions
Tagged with kinematics differentiation
191 questions
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
21
answers
5k
views
What happens when a car starts moving? The last moment the car is at rest versus the first moment the car moves
Imagine a car that's at rest and then it starts moving. Consider these two moments:
The last moment the car is at rest.
The first moment the car moves.
The question is: what happens between these 2 ...
- 371
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 $$x = 490t^2$...
13
votes
7
answers
3k
views
Can we divide a vector by another vector? How about this: $a = vdv/dx?$
My physics teacher told us that we can’t divide vectors, that vector division has no physical meaning or significance. How about this: $$a = vdv/dx.$$
It says acceleration vector equals velocity (as ...
- 876
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
11
votes
2
answers
3k
views
Kinematic equation as infinite sum
I'm not sure exactly how to phrase this question, but here it goes:
$v=\dfrac{dx}{dt}$ therefore $x=x_0+vt$
UNLESS there's an acceleration, in which case
$a=\dfrac{dv}{dt}$ therefore $x=x_0+v_0t+\...
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
10
votes
6
answers
3k
views
Physical intuition for higher order derivatives
Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
- 202
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
6
votes
2
answers
1k
views
Terminology for time derivative of speed (not velocity)
Is there any standard terminology for the derivative of the magnitude of velocity with respect to time (suitable for use in first-year Calculus)? The word ‘acceleration’, in its technical sense, is ...
- 201
6
votes
3
answers
1k
views
Is $\dfrac{dx}{dt}$ a fraction or not?
I am new to calculus and during my mathematics class my sir defined $\dfrac{dx}{dt}$ as $$dx/dt=\lim_{t\to t_1}\dfrac{f(t)-f(t_1)}{t-t_1}$$ and my sir made a clear statement that
$\dfrac{dx}{dt}$ ...
- 480
5
votes
2
answers
2k
views
How does instantaneous velocity or acceleration have any other numerical value than 0? [duplicate]
Instantaneous velocity is defined as the limit of average velocity as the time interval ∆t becomes infinitesimally small. Average velocity is defined as the change in position divided by the time ...
- 163
5
votes
4
answers
5k
views
How can there be really any instantaneous velocity?
I have read about Zeno's arrow paradox that tells us there is no motion of the arrow at a particular instant of its flight. It can be inferred that there can be no velocity at any instant. Moreover we ...
user36790
5
votes
3
answers
1k
views
Four velocity, acceleration, momentum and force in general relativity
I am getting slightly confused regarding the formal definitions of different four vectors in General Relativity. Many texts on relativity begin with four vectors and dynamics in Minkowski space, and ...
- 4,067
4
votes
3
answers
2k
views
What does "Just before" and "Just after" really mean in physics problems?
So I'm stuck in a dynamics problem that asks what is the acceleration of a body just after A, where A is the point that separates the motion of the body from a curvilinear path to projectile motion. ...
4
votes
6
answers
854
views
How to understand instantaneous velocity concept [duplicate]
When I started learning instantaneous velocity it didn't make sense to me. I don't understand in real life why we can't measure instantaneous velocity and therefore why we use this concept.
Or is this ...
- 311
4
votes
2
answers
312
views
Force and Accleration
It's just a basic question I had when I was studying physics years back,
So acceleration have two equations
$$a=\frac{F}{m}$$
and
$$a=\frac{\text{d}v}{\text{d}t}$$
So by the first equation, if I'm ...
- 171
4
votes
2
answers
261
views
The acceleration of circulation motion
We know that in circular motion the position vector is $r\hat{r}$. Then the velocity is the time derivative of it. So it gives $$dv/dt = r d\hat{r}/dt + \frac{dr}{dt} .\hat{r}.$$ now I can't ...
- 1,159
4
votes
2
answers
293
views
Do integrals of position make any sense? Do they have an application? [closed]
I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
- 41
4
votes
2
answers
18k
views
Why and when do we differentiate or integrate equations in physics? [closed]
I'm an engineering student and none of my professors ever explained why do we use derivations and/or integrations in physics. So I have this task, it goes like:
The object is moving in a positive ...
- 41
4
votes
2
answers
861
views
Integration of tangential acceleration with respect to time
Here, by tangential acceleration, I mean the component of acceleration along the velocity vector.
What do you get when you integrate tangential acceleration with respect to time? What does the '$v$' ...
- 1,106
3
votes
9
answers
4k
views
Can velocity be an undefined quantity?
We have the image below displaying the uniform velocity by time-distance graph. At every point velocity is constant but what if distance and time both become zero as at origin in the graph is? The ...
user66452
3
votes
3
answers
296
views
If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined? [closed]
If instant velocity at any given time $t_0$ is defined as the derivative of $x(t)$ at $t_0$, what if the derivative does not exist? How are we supposed to deal with $x(t)=|t|$ at $t=0$, or for more ...
- 39
3
votes
1
answer
271
views
How is $ \frac{dv}{ dt} = a $?
I know how , in the physical sense -
$$\frac {dv}{dt} = a$$
But, even after thinking a lot, I am not able to see the fault in this -
$$\frac {dv}{dt} = \frac {d(st^{-1})}{dt}
= \frac {sd(t^{-1})}...
- 141
3
votes
1
answer
133
views
Contradiction of a scalar product
Can anyone resolve this contradiction:
$$\vec{r}\cdot\dot{\vec{r}}=\frac{1}{2}\frac{d}{dt}\left(\vec{r}^2\right)=\frac{1}{2}\frac{d}{dt}\left(\left|\vec{r}\right|^2\right)\equiv\frac{1}{2}\frac{d}{dt}...
- 393
3
votes
1
answer
4k
views
What is the meaning of word 'rate' in physics?
Often, I have seen in physics the rate of change of velocity or something like that in kinematics. And in question based on speed, time and distance. I would like to know the meaning of the word rate ...
3
votes
3
answers
653
views
Confusion about successive derivatives of position in circular motion
Suppose we define a unit vector $\vec r$ along radial direction for a particle in uniform circular motion at an angular frequency $\omega$. Then we can write:
$$\vec r = \cos(\omega t)\hat i + \sin(\...
- 326
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
2
answers
133
views
Is this notation inconsistent? If not, can some explain why not?
Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states:
$y$ has a constant velocity of $10 \ \rm [m/s]$
$y=(0....
- 31
3
votes
1
answer
79
views
What do we get on differentiating the instantaneous displacement function?
I was doing kinematics when a silly question came to my mind. It is as follows:
When we differentiate $x(t)$ (position as a function of time), we get $v(t)$ (instantaneous velocity). Doing the reverse,...
- 137
3
votes
2
answers
160
views
Acceleration in terms of displacement
I am having problems understanding the derivation of acceleration in terms of displacement. The first step is fine:
$$a(x) = \frac{\mathrm dv(x)}{\mathrm dt}
= \frac{\mathrm dv(x)}{\mathrm dx} \frac{\...
- 131
2
votes
5
answers
346
views
Significance of $\frac{dv}{dx}=0$
Suppose an object is moving with varying acceleration in time.
What does it mean when it hits a point where $\frac{dv}{dx}=0$?
Does it mean the object has hit maximum velocity?
Assume the object ...
- 77
2
votes
4
answers
261
views
Show that $d\mathbf{v}^2/dt = 2\mathbf{v}\cdot d\mathbf{v}/dt$ using geometry only
I have just begun reading Modern Classical Physics by Thorne and Blandford and I am trying to wrap my head around their "geometric viewpoint" on classical mechanics. The first exercise in ...
- 510
2
votes
4
answers
733
views
Can a particle have no instantaneous velocity at all points of the path taken but a finite average velocity?
I have a question on kinematics.
Say the path traced by a particle is given by a Koch curve or Koch snowflake.
Now consider the particle starts from some arbitrary point $A$ on the curve and ...
- 4,984
2
votes
3
answers
193
views
Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?
My teacher has proved the following:
$$ \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
667
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
3
answers
131
views
Is motion in infinitesimal interval is linear?
As a kind of thought experiment I tried to think if a motion (including circular motion), when divided into infinitesimal time intervals is always linear motion (whether each interval of the motion is ...
- 121
2
votes
1
answer
2k
views
Meaning of time derivative of the Lorentz factor $\gamma$?
This question about the Lorentz factor $\gamma$ in special relativity. I know what $\gamma$ means and how to drive. I'm wondering if I have time derivative of $\gamma$, what dose it mean conceptually?
- 21
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
7
answers
44k
views
Is acceleration $a = s/t^2$, or $a = 2s/t^2$, or something third?
I'm having trouble understanding some of the stuff regarding movement in my introductory physics class (I never thought I'd say that...)
Acceleration is defined as $ a = \frac{s}{t^2}.$
Distance can ...
2
votes
1
answer
247
views
How to calculate jerk in uniform circular motion?
We can calculate the centripetal acceleration in circular motion by the equation v^2/r. But how do we calculate the jerk (which is acceleration over time)?
- 23
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
172
views
How does instantaneous velocity cause displacement in just one point? [closed]
I have a question.
Falling object graph is curve shape right?
And instantaneous velocity is tangent line but how does this velocity make displacement in distance? Because suppose instantaneous ...
- 311
2
votes
1
answer
291
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 \, .
$$
The particle traces a quarter arc of ...
- 131
2
votes
3
answers
179
views
Difference between $|d{\bf r}|$ and $d|{\bf r}|$
What is the difference between $|d{\bf r}|$ and $d|{\bf r}|$ and why are both of them not always equal to each other?
My question might seem stupid to some and will probably get downvoted but I have ...
- 733
2
votes
2
answers
105
views
Dimensions of a distance time relation
Recently I came accross a question which was:-
Suppose the velocity of a moving particle varies with time as
$$v=50t^2.$$
And we have to find out the acceleration at $t = 10s.$
I know that I can use ...