All Questions
Tagged with calculus newtonian-mechanics
119
questions
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What is the actual meaning of $dx$ in $W=-F.dx $, in work in thermodynamics?
what I want to ask is that the $dx$ in that formula is the displacement of piston or the displacement of the center of mass of the gas. also is there any situation where this clarity is useful.
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1
answer
103
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How to Find Trajectory of Particle?
Let’s say I have a particle, and I know all the forces acting on it at every position. (Let’s say the particle is in an electric/gravitational field to simplify the mathematics involved.) Now, is ...
3
votes
4
answers
193
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Is is true to say $F(x) = ma(x)$?
Considering the equation $F(t) = ma(t)$, I'm trying to figure out if the following is also always true:
$$F(x(t)) = m\cdot a(x(t))$$
I.e.: $F$ as a function of $X$ (the position, which itself is a ...
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43
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Why is time taken to go around the Sun to cover a small fixed angle proportional to the square of the distance?
I am reading Feynman's lost lecture. At this point, he asks us to consider points J, K, L and M which subtend equal angles at the sun S. And then he claims that triangles JKS and KLS are similar ...
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44
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Energy Dissipated by Damper Infinitesimal Derivation
If we consider a damper (dashpot) element that exerts a force opposite the direction of motion proportional to the velocity, i.e. $$ \vec{F} = -c \vec{v}$$
Therefore, we can consider an infinitesimal ...
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1
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84
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Understanding the double integral solution to Newtons second law?
I was following this lecture on Newtons Laws.
https://youtu.be/2tHpgQmnH3A?si=Wbp36oBS_4b1HhIi
At 31:56 in the video, the board has a very general solution to Newton's second law.
However the second ...
0
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3
answers
185
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What does it mean in terms of energy if power is increasing with time? [closed]
Recently I have been studying work and energy and in this chapter I encountered with the term power. In terms of work power is written as $dW/dt$. So I have a doubt that suppose power is increasing. ...
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1
answer
94
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What is $V$ in $a$=$V$$dv$/$dx$? [duplicate]
$a$=instantaneous acceleration
$V$=instantaneous velocity
$x$=position
$dx$=small Chang in position
$a$=$dv$/$dt$
multiplying numerator and denominator by $dx$,we get
$a$=$dv$.$dx$/$dx$.$dt$
now we ...
- 161
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2
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88
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Equilibrium of a body with potential energy as a function of position
We know that if the potential energy of a body, say $U(x)$ of a body is known as a function of its x-coordinate, for equilibrium, $$\frac{dU(x)}{dx} = 0$$ Also, several sources suggest that for the ...
- 301
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66
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Can someone help me with differential equation please? [duplicate]
here is the topic of the problem:
You are given $2$ baseballs (consider them as perfect solid spheres) have equal properties with mass $m = 0,142kg$, radius $r_0 = 0.037m$ in the space and thay are $...
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30
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The choice of the direction of the displacement vector when calculating potential energy of a system
Here, when referring to potential energy, I will take gravitational potential energy as an example. Consider the following diagram where two point masses $m_1$ and $m_2$ at a distance $r$ from each ...
-1
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2
answers
237
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(Physics 2, Waves) Why does $\tan(\theta) = dy/dx$? [closed]
In the following example:
At the very last step, how does the author get that $\tan(\theta) = dy/dx$? To which $dy$ and $dx$ is this referring to? It can't be the same $dx$ that is labelled in the ...
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2
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52
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Work-Energy Theorem for a path that is not smooth
In the analysis of Newtonian Mechanics for a single particle, we come across the definition of work and also the Work-Kinetic Energy theorem:
For a single particle, the work done on a particle by a ...
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160
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How do I write the gradient in angular coordinates ($\theta_1$, $\theta_2$, $\theta_3$)?
I have to find $\tau$ by finding the gradient of $U(\theta_1, \theta_2, \theta_3)$, where my coordinates are $(\theta_1, \theta_2, \theta_3)$. I assume the gradient is not the simple Cartesian ...
2
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1
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Why is the gravitational potential of a uniform disc not symmetric about its center?
Consider a uniform, infinitely thin disc of surface mass density $\sigma$ and radius $R$ placed in the $xy$-plane with its center as the origin.
The gravitational potential at a point on the axis of ...
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1
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1
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65
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Conservation Principle
We are introduced to Principle of Conservation of Linear Momentum via the Newton's Second Law
$$\vec{F_{net}}=\frac{d\vec{p}}{dt}$$
It states when net external force equals zero then $\vec{p}=$...
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4
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4
answers
2k
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Help me understand the derivation of the kinetic energy formula please
In my physics textbook, kinetic energy is defined as $W$Net $=$ $\int m\frac {dv}{dt}$ $dx$ This makes sense to me just fine. The book goes on to rearrange the integral to say the following:
$W$Net $=...
0
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2
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268
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Circular motion equivalent in three dimensions [closed]
Are there equations or even a concept of circular motion/tangential acceleration/centripetal acceleration in three dimensions? Maybe something called "spherical acceleration"? or am I just ...
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360
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Is the acceleration vector half of the gradient of velocity squared?
Consider the differentiation of speed squared with respect to time:
$$\frac{d(v^2)}{dt}=\frac{d(\mathbf v\cdot\mathbf v)}{dt}$$
$$=2\mathbf v\cdot\frac{d\mathbf v}{dt}$$
$$=2\mathbf v\cdot\mathbf a$$
$...
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6
answers
116
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Deriving Work-Kinetic Energy Theorem
I am currently reading Physics for Scientists and Engineers (Ninth Edition) by Serway and Jewett and in Chapter 7.5, a derivation of the work-kinetic energy theorem was shown.
To give context, ...
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85
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2D rotation dynamics/control systems as a complex number
I have a dynamic system (it's a rocket in a 2D plane), that I'd like to model the orientation of using complex numbers to remove the need for trig functions in my ode.
I'm having trouble defining the ...
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15
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3
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Why does solving the differential equation for circular motion lead to an illogical result?
In uniform circular motion, acceleration is expressed by the equation
$$a = \frac{v^2}{r}. $$
But this is a differential equation and solving it gets the result $$v = -\frac{r}{c+t}.$$
This doesn’t ...
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2
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291
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Finding angular frequency via integration of Newton's Second Law for a physical pendulum
For context: I am a student enrolled in AP Physics C with prior knowledge from AP Calculus AB and AP Physics 1.
We just collected data for a lab to determine an experimental value for g. The setup ...
1
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1
answer
523
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Acceleration as a function of displacement
I am given a question such that a 0.280kg object has a displacement (in meters) of $x=5t^3-8t^2-30t$. I need to find the average net power input from the interval of $t=2.0s$ to $t=4.0s$.
I know the ...
- 115
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0
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12
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Issue with work vs force for calculating spring constant [duplicate]
Lets say I have a spring with spring constant k. I put a 10kg weight on the spring and it compresses the spring one meter before stopping. We know that at this point the downwards force is equal to ...
3
votes
5
answers
2k
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Guidelines to calculate moment of inertia
The moment of inertia is defined as
$$I = \int r^2 dm$$
but I am not sure how to proceed with solving the above integral. All examples I have seen seem to be done with different strategies. They ...
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2
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178
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How do we show that the work done by a variable force (in one dimension) is the area under the $F$ vs. $x$ curve?
In my physics textbook, to show that work is the area under the $F$ vs. $x$ curve, the author first writes the relation $dw = F dx$. This part makes sense to me. From there, the author writes, $$W = \...
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1
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1
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256
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Tidal forces mathematics
Let's calculate the difference in force, $\Delta F$, experienced by
the rocks. Because $\Delta r$ is very small compared to $r$,
$$\Delta F = F_{\text{out}} - F_{\text{in}} \approx\frac{dF}{dr}\Delta ...
0
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0
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126
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Work-Energy Principle Derivation
I am currently in Mechanics I and both my professor and my book have derived the work principle in this way and I even asked about its derivation during class, but it has me puzzled.
I don't ...
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2
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83
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Kinematics confusion regarding sign of integration
I was solving some problems regarding non-inertial frames, and Newtonian mechanics in general, when I faced a major doubt regarding one of the seemingly simple topics, and I'd appreciate it if someone ...
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1
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81
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Doubt in finding center of mass of extended bodies [closed]
While finding center of mass of extended bodies, we generally write the coordinate of the starting point of the infinitesimal mass element. Why not the ending or starting point? How does the error ...
0
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1
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39
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Velocity while falling from table
Suppose there is a chain of mass $m$ in a semicircular tube of radius $r$. If it's pushed, then find the velocity of the chain while leaving the tube.
This is a simple problem which we can do using ...
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2
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4
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633
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Work done by a vector field (Force field) on a particle travelling along a curve
Assume a particle travelling along a curve, the work done by any Force field on the particle while moving along a curve is given by the line integral of $\vec{\bf{F}} \cdot \vec{\bf{dr}}$, but shouldn'...
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1
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1
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129
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Finding velocity $v$ and position $r$, given a time $t$ under the acceleration of a gravitational force [closed]
I was messing with the maths, when I tried to find the velocity as a function of time, $v(t)$, and the position, also, as a function of time, $r(t)$ under the gravity force.
$$ m \ddot{r} = -G \frac{...
- 1,515
2
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1
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67
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Forces along and perpendicular to a curve
A uniform rope of length $l$ is suspended from two hinges, making an angle of $\theta$ with the horizontal at the hinges. Find the depth $d$ of the lowest point of the rope.
Similar questions include ...
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1
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166
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Maximum height of a projectile when $g$ is not constant [closed]
How can I calculate the maximum height of a projectile that is launched from the surface of the earth with a given initial velocity? (ignoring air resistance in the atmosphere)
I understand how to ...
26
votes
14
answers
4k
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Explaining how we cannot account for changing acceleration questions without calculus
For context, I am a high school physics teacher.
I am teaching students about the basics of electromagnetic force between two point charges. The equation we use is $F=\frac{kq_1q_2}{r^2}$.
This gives ...
1
vote
2
answers
73
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How do you differentiate this differential equation? [closed]
I have to differentiate this equation (Gravitational force between N-Bodies)
$\begin{align}
\frac{d^2}{dt^2}\vec{r_i}(t)=G
\sum_{k=1}^{n}
\frac
{m_k(\vec{r}_k(t)-\vec{r}_i(t))}
{\lvert\...
0
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1
answer
25
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Issue with a derivation in Marion's Dynamics [closed]
I was solving problem 2-14 in Marion's "Classical dynamics of particles and systems" edition 5. In this problem we calculate the range of a trajectory to be $d=\frac{2{v_0}^2\cos{\alpha}\sin{...
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3
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181
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Trouble understanding the center of mass equation
I'm learning about center of mass, but I have trouble understanding the definition. How is $x_{com}=\frac{1}{M}\sum_{i=1}^{n}m_ix_i$ equal to $x_{com}=\frac{1}{M}\int xdm$?
At first I thought it ...
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1
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2
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81
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Is it possible to lift an object from rest with constant power?
This is inspired by the following question.
Consider some object which I want to lift from rest with a constant power throughout the whole process; the power I apply when lifting the object from rest ...
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3
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165
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Power and work contradiction
A body is starting from rest. A force is acting on it for a short period of time. In that given time, power delivered to it at any instance $t$ is given
$$P = F \cdot v_1 = ma \cdot v_1 = mv_1^2/t,$$
...
1
vote
2
answers
369
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Area Swept by particle under central forces is an approximation
From Kepler's second law, we infer, the conservation of angular momentum is equivalent to saying the areal velocity is constant,
And the proof goes like this
$$ mr^2{\dot\theta=L}
$$
where $L$ is ...
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2
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1k
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Why distance equals initial velocity times time Plus acceleration over two times time Squared? [duplicate]
i am a beginner in physics and I do not understand why is the d=vi(t)+(1/2)a(t^2).
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-1
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2
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92
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Infinitesimals meaning in this context
I was solving this rocket propulsion's classic mechanics exercise: M is the instantaneous rocket's mass, and v its velocity. The exhaust gases are ejected with speed 𝑢 relative to the speed 𝑣 of the ...
- 169
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1
answer
33
views
Average torque on a Projectile of mass $m$ with initial speed $u$ and angle of projection $θ$ between initial ($P$) and final ($Q$) positions is [closed]
Question is as follows:
Average torque on a Projectile of mass $m$ with initial speed $u$ and angle of projection $θ$ between initial $(P)$ and final $(Q)$ positions is
I researched a lot but wherever ...
0
votes
0
answers
142
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How to calculate the derivative of the angular momentum vector $ d\vec L = d(\hat I \vec \omega)?$
My last question, but also the most important one How to calculate the derivative of the angular momentum vector?
$$ d\vec L = d(\hat I \vec \omega)$$
I'm especially interested in derivative tensor to ...
0
votes
1
answer
120
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Why can we multiply by $dt/dt$ to change variable of integration? Please look at equation 5-20 [closed]
I am struggling to understand why can we just multiply by $dt/dt$. I was thinking it was just a change of variables, but I cannot come up how that works. Can someone explain why we are allowed to do ...
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0
votes
3
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147
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The force of gravity between a shperical shell and a particle
I am trying to understand the proof of why the force acting on a spherical shell and a particle is
$$\frac{GMm}{r^2}$$
Where M is the mass of the sphere and m is the mass of the particle.
I am looking ...
- 5
1
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3
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65
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Position Dependence in Equation of Motion
Our lecturer gives study material which contained that Newton's second law could be written as:
$$ \begin{aligned} F &= m \ddot{x} \\ &= m \frac{d \dot{x}}{dt} \\ &= m \frac{dx}{dx} \frac{...
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