All Questions
Tagged with kinematics differentiation
191 questions
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In Radius of Curvature calculation why do I have to assume $\text{d}^2x/\text{d}t^2=0$?
Recently, I was calculating the radius of curvature of projectile trajectories at certain points. There are two ways to do the same:
Given the velocity and acceleration of the particle at some point, ...
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55
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What do you call $ \frac{d^2 r}{dt^2}$ in polar coordinates? [duplicate]
In polar coordinates, one finds centripetal acceleration as:
$$ a_c = \frac{d^2 r}{dt^2}- \frac{v^2}{r}$$
Where $|r|$ is distance from center to particle, $v$ is tangential velocity.
My question is ...
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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
$$\frac{d x_k(q_1,q_2,...
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46
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1/velocity for higher dimensions
I have a somewhat basic question. I am sorry if it trivial.
Denote the velocity by $v=\frac{dx}{dt}$ suppose that $x \in \mathbb{R}^n$ and I want to parametrize $t$ in $x$ and compute $\frac{dt}{dx}$. ...
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152
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Product rule for 4-vectors and derivation of 4-force form
In deriving the form for the 4-force in special relativity, we begin with
$$\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 $...
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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 ...
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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\...
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46
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Name for the set Displacement, Velocity, Acceleration, etc
Is there a name for the set Displacement, Velocity, Acceleration, Jerk, etc?
The only name I can think of is 'Derivatives of displacement (wrt time)' which is rather long.
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2
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84
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Acceleration in a non-inertial reference frome - derevation
The general velocity equation for a point B in on body rotating and translating about point A with respect to the inertial reference frame say 'xyzo' can be expressed as,
$\vec{r_{B/o}} = \vec{r_{A/o}...
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1
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59
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From infinitesimal momentum volume to infinitesimal rapidity & tranverse momentum
I'm trying to derive a relationship given in a paper which is used to obtained a differential cross-section distribution in function of rapidity and transverse momentum of final state particles,
$$
\...
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2k
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Derivation of centripetal acceleration
While reading HC Verma chapter 7 circular motion I came across a derivation which I couldnt understand. I have marked my doubt with red. I don't understand from where +dw/dt [- i sine +j cos0] came ...
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3
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960
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Acceleration derivative
I am reading Morris Kline's "Calculus" and I fail to understand this notation:
We have acceleration to which an object $r$ feet from the center of the earth (and above the earth) is subject. If we ...
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2
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158
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What does it mean for velocity to be a derivative of position, if position a point, not a function? [closed]
So in mass-spring simulation I encountered that one simulates particles by using positions and velocities of particles etc.
People may say that velocity is the derivative of position.
But isn't "...
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1
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106
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Radius as a constant while deriving of formula for acceleration in circular motion [closed]
While deriving acceleration in circular motion, we differentiate $\vec{v}=\vec{\omega}\times\vec{r}$
Here we differentiate by product rule and write $\frac{dr}{dt}$ as $v$.
So we know $\vec{s} =\vec{\...
-1
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1
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1k
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What is the physical interpretation of a differential equation? [closed]
I would like to learn more about differential equations and their interpretation. I know the derivation rules, but I fail big time in interpreting and understanding the functionality of them. For this,...
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1
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762
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Uncertainty in Range of Projectile [closed]
If we are given that a projectile is launched with velocity 10m/s at an angle of $45^\circ$ and uncertainty in angle is of $0.5^\circ$ . What is the uncertainty in the range of projectile.
The problem ...
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2
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80
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Problem with resources, Walter Lewin's third lecture
I've watched Walter's third lecture in 8.01 and I have a small problem with the last part, where he says that $$\vec r_t=x_t\cdot \hat x\ +\ y_t\cdot \hat y\ +\ z_t\cdot \hat z \\ \vec v_t=\frac{d\vec ...
-1
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1
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164
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Given a Postion-time curve/function, how do I find the time spent per unit position?
I have recordings of the position time curve for a given 1D actuator.
I'm trying to find out the time spent per unit length.
To get this relationship, I tried to take an example of a linear function:
$...
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2
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67
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Instantanous and uniform velocity and acceleration [closed]
If the mathemical expression of instantanous velocity is $d/t$, what is the mathematical expression of uniform velocity.
If the mathematical expression of instantanous acceleration is $v/t$, what is ...
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1
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51
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Assistance interpreting equation
Given a position function of a particle:
$$
\mathbf r=r\,\hat{\mathbf r}\left(\theta\right),
$$
where $\hat{\mathbf r}(θ)$ is the polar unit vector, to find the velocity, we take the derivative which ...
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1
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How to find Net Force with constant velocity? [closed]
Does having a constant velocity always make the acceleration equal zero?
For example: A 5 kg ball is moving at constant velocity of 15 m/s. What is the net force on the ball?
If the formula is $F_{...
-1
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1
answer
240
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Why is velocity mathematically describes as a division? [duplicate]
I want to know why, in kinematics, is velocity described as $v = \frac{\Delta x}{\Delta t}$, and why it is not described as any other expression (like a multiplication), why does a division is the one ...
-1
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2
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273
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How can I show that the acceleration vector for uniform circular motion undergoes uniform rotation?
Does it suffice to show that the dot product between the acceleration vector and the derivative of the acceleration vector = 0?
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2
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368
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How instantaneous speed is defined as magnitude of instant velocity? [closed]
Let $s=$distance (a variable)
we define instantaneous speed = magnitude $\left[\frac{ds}{dt}\right]$.
However instantaneous speed is also defined as magnitude of instantaneous velocity
i.e. ...
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1
answer
61
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Simple 2D motion vectors [closed]
I am curious if the initial velocity of $x(t)=-3-4t+2t^2$ can be calculated from only this given in another way than just differentiation, by using the constant acceleration formulas perhaps?
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2
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121
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Acceleration and velocity
I'm a freshmen student, I got this question in my mind why we consider acceleration based on velocity not speed.
as far as I know, velocity will be zero if we go and back from A to B although speed ...
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2
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89
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For how long is an objects velocity it's instantaneous velocity at time $t$?
Basically I'm asking if an object's instantaneous velocity at time $t$ is $8m/s$ and its instantaneous velocity at time $t^+$ (idk latex, but basically the t + an infinitely small number) is $10m/s$, ...
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4
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213
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Why the need for defining the velocity as a derivative? [closed]
Something intuitive and fundamental as the concept of velocity (of a particle for example) in classical physics is defined as a derivative, a concept to me quite vague and strange, although i know its ...
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2
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122
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Why does $\vec{a}=\vec{\omega}\times \vec{r}$ as well as the velocity does?
Today I came in class and in one of the problems the teacher used $\vec{a}=\vec{\omega}\times \vec{r}$ which made me very confused because I don't know where it comes from, it seems pulled out of thin ...
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1
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59
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Need help in understanding Tangential Acceleration [closed]
I am studying Circular motion and I am confused about tangential acceleration and tangential velocity. I am studying uniform circular motion and it says the tangential acceleration is $0$ in uniform ...
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2
answers
113
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If an object is travelling at say 35 mps, if I somehow stop the time, is the speed zero or 35 mps? [closed]
I know we only go close to zero, not equal zero, but if that somehow happens, will it be zero as it is at rest or 35mps as when you resume the time, the speed is 35mps?
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3
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96
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Why is it wrong to find centripetal acceleration using change of velocity over change of time?
This question asks to find the centripetal acceleration by giving the initial and final velocity over the change of time.
As shown, my book combined two rules to find the acceleration. I utterly ...
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1
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91
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From where does the expression of the tangential accerelation come from?
I've seen so many times that the expression of the tangential acceleration is known to be: $$a_t=\ddot{s}$$ but from the expression of the acceleration in spherical coordinates, in the tangential ...
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1
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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 ...
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1
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122
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How can there be instantaneous rate of change? [duplicate]
To find rate of change you need two instants.
how is the rate of change calculated at a particular instant when at least two instants are needed to find it?
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0
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70
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Use of $dv/ds$ in defining acceleration [duplicate]
We can write acceleration as either
$dv/dt$ or $v dv/ds$.
And surprisingly the work-energy theorem arrives from the second definition.
I feel it would be fundamentally understanding towards work ...
-3
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2
answers
290
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Kinematics problem invloving position and time [closed]
An object is moving along X axis with position as a function of time given by $x = x(t)$. Point $O$ is at $x = 0$. The object is definitely moving towards $O$ when
1. $\mathrm dx/\mathrm dt < 0$
...
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2
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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 ...
-4
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1
answer
74
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Find out which coordinate changes at faster rate [closed]
Suppose we have a particle, which moves along a path (in x-y plane) and say its path is the curve, $ 12y = x^3 $ .
I need to find out which coordinate (x or y) changes at faster rate at any given ...
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1
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97
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Differentiation [closed]
Why is
$$\frac{d}{dt}v^2=2v\frac{dv}{dt},$$
When:
$$\frac{d}{dx}x^2=2x,$$
where $v$ is velocity? I don't understand why the variable $x^2$ has the derivative of $2x$, whereas the variable velocity has ...
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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 $ = $m \frac{d}{dt} (v \cdot v)/2$?
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