All Questions
Tagged with kinematics differentiation
191 questions
0
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2
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353
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Why isn't tangential acceleration just always 0?
This is probably a very stupid question but I can't help me.
Tangential acceleration is $\vec{a_t}=\frac{dv}{dt}\frac{\vec{v}}{v}=\frac{\vec{v} \cdot \vec{a}}{v} \frac{\vec{v}}{v}$. Since $\vec{a}$ is ...
1
vote
1
answer
80
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On the derivation of the jerk term in tensor notation
The jerk vector $\mathbf{J}(t)$ may be (element-wise) written as
$$J^{i} = \frac{d A^{i}}{d t} + \Gamma^{i}_{jk} A^{j}V^{k}~~~~(*).$$
However, I can't correctly get the second term of the RHS above. ...
0
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0
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85
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Cartesian coordinate velocity and generalized coordinate velocity
use $x_k$ to denote the kth component of cartesian coordinate, and $q_k$ to denote the generalized coordinate.
Taking the derivate of $x_k(q_1,q_2,q_3,t)$ w.r.t. time, we have
$$\frac{d x_k(q_1,q_2,...
0
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0
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46
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1/velocity for higher dimensions
I have a somewhat basic question. I am sorry if it trivial.
Denote the velocity by $v=\frac{dx}{dt}$ suppose that $x \in \mathbb{R}^n$ and I want to parametrize $t$ in $x$ and compute $\frac{dt}{dx}$. ...
1
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2
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159
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One object moves along the cycloid at a constant rate, how about its acceleration? [closed]
We know that the parametric equation:
$$x=R(\theta+\sin(\theta))$$
$$y=-R(1+\cos(\theta))$$
and the constant velocity $c$.
How do I prove that the acceleration of the object in the $y$ direction is ...
0
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1
answer
129
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Why intuitively is the tangent vector the derivative of velocity of position with respect to their modulus?
When trying to find the tangential velocity, many textbooks define the tangent direction as one of the following:
or
Intuitively, why is the tangent vector the derivative of the position with ...
-2
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1
answer
49
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What does the derivative of tangent means? [closed]
While studying the circular motion I had to find the derivative of a tangent so I thought what the derivative of a tangent could probably mean since the derivative of position gives velocity.
Or think ...
0
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1
answer
60
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A particle moves along the curve $y=mx^a + c$ with a constant speed $v$. Find the velocity at x=n (in $i cap, j cap$) and acceleration at x=n [closed]
A particle moves along the curve $y=mx^a + c$ with a constant speed $v$. Find the velocity at x=n (in $i cap, j cap$) and acceleration at x=n.
I tried this question as:
I found out the slope of curve ...
0
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1
answer
42
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Is such a situation realistically possible where $v$-$t$ graph is continuous but $a$-$t$ graph is not?
Taking for example $v = \cos(t-1)$ from $t \in [0,1]$ and $v = e^{t-1}$ from $t \in (1,\infty)$ and $t \ge 0$. At $t = 1$, the function shifts from cosine to exponential, but remains continuous since ...
0
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1
answer
53
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Differentiation/Integration [closed]
I am given $v=6-4x$ where $v$ is velocity and $x$ is position of the body. I am now asked to find displacement between $t=1.4$, distance covered from $t=1.4$, average velocity from $t=2.5$, average ...
1
vote
3
answers
307
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What does the derivative of unit vector of velocity with respect to time represent?
let an object move with a constant accelration a. in my book,the following derivatve is said to be non-constant(variable).
$$\frac{d[\frac{v}{|v|}]}{dt}$$
what does this mean?
as far as i can think,it ...
3
votes
3
answers
296
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If the displacement of an object is not differentiable at some point, say $x(t)=t\sin(1/t)$ at $t=0$, how is its instant $v$ defined? [closed]
If instant velocity at any given time $t_0$ is defined as the derivative of $x(t)$ at $t_0$, what if the derivative does not exist? How are we supposed to deal with $x(t)=|t|$ at $t=0$, or for more ...
3
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2
answers
233
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Generalization of straight line motion under constant acceleration
My question is that, we all know the three equations of straight line motion under constant acceleration,
\begin{align}
x & =x_{\rm o}+v_{\rm o}\,t+\tfrac12 \mathrm a\,t^2
\tag{1d-a}\label{1d-a}\\
...
1
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1
answer
458
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Expressing acceleration in terms of velocity and derivative of velocity with respect to position
we know that
$$a = \dfrac{dv}{dt}$$
dividing numerator and denominator by $dx$, we get $$a=v\dfrac{dv}{dx}$$ provided that $dx$ is not equal to zero or instantaneous velocity not equal to zero
when I ...
2
votes
3
answers
131
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Is motion in infinitesimal interval is linear?
As a kind of thought experiment I tried to think if a motion (including circular motion), when divided into infinitesimal time intervals is always linear motion (whether each interval of the motion is ...
-4
votes
1
answer
71
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Given that $m \dot v \cdot v = 0$ , how is it equal to $m \frac{d}{dt} (v \cdot v)/2$? [closed]
While studying about scalar triple product in vector algebra, I stumbled upon the following question with the solution.
I want know how is $m \dot v \cdot v $ = $m \frac{d}{dt} (v \cdot v)/2$?
0
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1
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78
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Particle paths - the distance moved by a particle in a velocity field
This question is is context to particle paths.
Particle paths are trajectories of a given particle in the velocity field:
$$\boldsymbol{u}(\boldsymbol{x},t)$$
A particle location at position $\...
2
votes
1
answer
291
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Is the relation "slope=velocity" mathematically valid?
$\text{Slope= tan(angle with respect to positive X-axis)= scalar output}$
$\text{velocity= a vector }$
Source: Hugh D Young_ Roger A Freedman - University Physics with Modern Physics In SI Units (...
2
votes
1
answer
267
views
Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?
Consider a particle that moves around the coordinate grid. After $t$ seconds, it has the position
$$
S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, .
$$
The particle traces a quarter arc of ...
3
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1
answer
4k
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What is the meaning of word 'rate' in physics?
Often, I have seen in physics the rate of change of velocity or something like that in kinematics. And in question based on speed, time and distance. I would like to know the meaning of the word rate ...
3
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2
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133
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Is this notation inconsistent? If not, can some explain why not?
Im working through a textbook section on particle kinematics. An example given is relating vertical velocity to horizontal velocity and states:
$y$ has a constant velocity of $10 \ \rm [m/s]$
$y=(0....
0
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2
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71
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Why quantities in physics are always talking about rates? [closed]
I get the idea that physics wishes to study changes to discover new rules.
But why is everything related to rates? Acceleration,Velocity?
Could we use something else apart from these?
What can you ...
0
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2
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139
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How do I find the unit vector for the tangent? [closed]
A body moves in the track from A to B to C and so on. The magnitude of the velocity in any given moment is
$|\vec V|$=$t^3$$[m/s]$. The body arrives to point B at $t$$=$$2$$[s]$. The line in the draw ...
0
votes
1
answer
225
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The time derivative of a vector not defined in terms of the time variable $t$
Recently I got a question where I needed to determine the time derivative of a position vector. However, the vector didn’t have the variable $t$ but instead had $x$, $y$, and $z$ as its terms, so I ...
2
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3
answers
193
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Is $ d \mathbf v · d \mathbf v = d \mathit v^2 $?
My teacher has proved the following:
$$ \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} $$
Because ...
4
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3
answers
2k
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What does "Just before" and "Just after" really mean in physics problems?
So I'm stuck in a dynamics problem that asks what is the acceleration of a body just after A, where A is the point that separates the motion of the body from a curvilinear path to projectile motion. ...
3
votes
3
answers
653
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Confusion about successive derivatives of position in circular motion
Suppose we define a unit vector $\vec r$ along radial direction for a particle in uniform circular motion at an angular frequency $\omega$. Then we can write:
$$\vec r = \cos(\omega t)\hat i + \sin(\...
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 ...
4
votes
2
answers
312
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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 ...
34
votes
7
answers
5k
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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 ...
-3
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2
answers
308
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If we divide the second equation of motion by time $t$, why don't we get the first equation of motion where has $1/2$ come from? [duplicate]
The first equation of motion is $v = u + at$.
The second equation of motion is $s = ut + \frac{at^2}{2}$.
If we divide the second equation of motion by time $t$, why don't we get the first equation of ...
2
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4
answers
667
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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'...
1
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2
answers
167
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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 ...
1
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5
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385
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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 ...
0
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3
answers
232
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Are acceleration and velocity simultaneous? [closed]
I would think yes because, if a rope tied to a swinging rock breaks, the rock flies off in the direction that is perpendicular to the direction of the last instant of the acceleration. The ...
0
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1
answer
88
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Does "$\rm m/s$" mean the same thing when used for instantaneous velocity and for mean velocity?
The following question on Philosophy SE https://philosophy.stackexchange.com/q/73366/ relies on this "given"
"suppose that the instantaneous velocity of object A is $1$m/s and that the mean ...
1
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2
answers
557
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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
0
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1
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468
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Partial derivative of a 4-velocity
Trying to do some basic manipulations with 4-vectors and I have a question about the proper (no pun intended) approach. It's probably easiest if we look at a simple example. So let's define a 4-...
9
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4
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2k
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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 ...
0
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0
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152
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Product rule for 4-vectors and derivation of 4-force form
In deriving the form for the 4-force in special relativity, we begin with
$$\frac{d}{d\tau}p^{\alpha}=m\frac{d}{d\tau}u^{\alpha}=f^\alpha$$
where $\tau$ is the proper time, m is rest mass.
Since $...
1
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4
answers
3k
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What is proper time, proper velocity and proper acceleration?
I am trying to derive the relativistic rocket equations found here [(4),(5),(6),(7),(8)] but I do not understand proper time, proper velocity and proper acceleration.
Define a point $P$ with ...
1
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4
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58
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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 ...
0
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1
answer
71
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Derivative of four-vectors [closed]
I have to take the derivative of $(P+2k-2q)^2$ with respect to $k^{\alpha}$. What I am doing is writing
$(P+2k-2q)^2=(P+2k-2q)^{\alpha}(P+2k-2q)_{\alpha}$
Then its derivative comes to be
$2(P+2k-...
0
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3
answers
511
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Can you use $a=$$\frac{\Delta v}{\Delta t}$ instead of $\frac{dv}{dt}$ to find instantaneous acceleration?
Can you use $\frac{\Delta v}{\Delta t}$ instead of $\frac{dv}{dt}$ to find instantaneous acceleration?
0
votes
1
answer
87
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What mean this momentum-derivative?
I'm working with quantum gravity. I have to make a Taylor-series. I got some help for this, but I have problem with understanding the formalism. So, I have the operator $A((P-p)^2)$, which needs to ...
0
votes
1
answer
347
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Projectile motion - differentiating the equation of trajectory to find the maximum height
I have the equation of trajectory:
$ y = x\tan \theta - {\displaystyle gx^2 \over \displaystyle2u^2\cos^2 \theta}$
I also know that the maximum height is given by:
${\displaystyle u^2 \over\...
0
votes
2
answers
299
views
Velocity as a property
Is velocity considered to be a property like mass and weight that can be measured at a single moment in time, such as mass of X measured at time T1, or is it a property that needs to be measured over ...
1
vote
2
answers
220
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Average velocity and instantaneous velocity
In some books of Physics in Italian language, they write that the instantaneous velocity $v$, is:
$$v=\frac{dr}{dt}=\lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t}$$
where $v_{\text{avg}}={\Delta r}/...
1
vote
2
answers
133
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Related to the information contained in $a = v \frac {dv}{ds}$
While studying kinematics I came to the definition of acceleration which is $a = \frac {dv}{dt}$. But from this equation we can derive that $ a = v \frac {dv}{ds} $ which when I evaluate at $v=0ms^{-1}...
11
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4
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3k
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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, ...