64
votes
Accepted
Why do we need instantaneous speed?
Because instantaneous speed affects physics. Imagine a wall $10~\textrm m$ in front of you. You walk towards it smoothly over a timeframe of, say, $20~\textrm s$, and without getting slower, you walk ...
- 2,703
49
votes
Accepted
How is it possible to differentiate or integrate with respect to discrete time or space?
Let's say space is really a lattice with spacing $\Delta x$. It turns out that this idea has more trouble with experiment than you might think, but we can plow ahead for the purposes of this question.
...
- 52.3k
43
votes
Does it make sense to take an infinitesimal volume of shape other than a cube?
Infinitesimal volume elements do not have to be cubes.
Some familiar examples come from typical solids of revolution problems from calculus 1/2. Typically one discusses using either the "disk/...
- 58.2k
42
votes
Why use Fourier series instead of Taylor?
The complex exponentials are eigenfunctions of the derivative and integral operators. So if you're analyzing linear differential equations, and using Fourier series, then you can consider each term on ...
- 28.8k
41
votes
How does instantaneous velocity or acceleration have any other numerical value than 0?
Suppose you are travelling at a uniform velocity and you cover 1 meter in 1 second. Your average velocity is
$$\frac{1\ {\rm m}}{1\ {\rm s}} = 1 \frac{\rm m}{\rm s}.$$
If you consider a 1 ...
- 28.8k
39
votes
What's the difference between average velocity and instantaneous velocity?
Your question is legitimate and I don't understand why it got downvoted.
The confusion arises in the difference between average and instantaneous velocity.
Consider this example: a car moves at 10 m/...
- 4,746
38
votes
Accepted
How does instantaneous velocity or acceleration have any other numerical value than 0?
$$v_\text{average}=\frac{\Delta s}{\Delta t}$$
$$v_\text{instantaneous}=\lim_{\Delta t\to0}\frac{\Delta s}{\Delta t}$$
If the time interval gets infinitesimally small $\Delta t\to 0$, then you are ...
- 51.7k
31
votes
Accepted
Why does solving the differential equation for circular motion lead to an illogical result?
That equation is misleading.
You wrongly assume that $a$ is $dv/dt$ here, which it isn't.
Let us first rewrite the centripetal acceleration equation properly:
$$|\vec {\mathbf a}_N| = \dfrac {\left|\...
- 6,631
30
votes
Derive vector gradient in spherical coordinates from first principles
You asked for a proof from "first principles". So let's do it. I'll highlight the most common sources of errors and I'll show an alternative proof later that doesn't require any knowledge of tensor ...
- 1,954
30
votes
Accepted
Does the logarithm of a non-dimensionless quantity make any sense?
In equation (7) you have the expression
$$−P\ln(P−Rv(n+1))+P\ln(P)$$
But since this a difference between two logarithms
you can rewrite the expression
(remember $\ln a - \ln b = \ln \frac ab$) as
$$...
- 40.3k
30
votes
Why do we need instantaneous speed?
It is really simple:
Average speed is as good as instantaneous speed only if speed does not change with time. If you are studying a body with rapidly changing speed then using average speed to ...
- 4,565
27
votes
Explaining how we cannot account for changing acceleration questions without calculus
You do, in fact, have to take into account the change in separation distance between charged objects when analyzing the dynamics of the system. This is done mathematically through the use of ...
- 1,004
25
votes
Accepted
When the direction of a movement changes, is the object at rest at some time?
After the invention of modern calculus and notions like continuity and differentiability, the answer is quite trivial in Newton's formulation of mechanics assuming the body is moving along a line. ...
- 75.6k
21
votes
Accepted
Why do we equate an indefinite integral to a specific value?
In physics we frequently leave off the limits of the integral when the limits can be figured out from the context. So, in the first case, the actual relation is:
$$x(t) = x(0) + \int_0^t \dot{x} \...
- 22.7k
21
votes
Does it make sense to take an infinitesimal volume of shape other than a cube?
Your comments (and to a lesser extent, your Question) indicate a severe confusion about ever having an infinitesimal volume. You never construct an infinitesimal volume. Infinitesimal volumes appear ...
- 1,777
21
votes
Accepted
Explaining how we cannot account for changing acceleration questions without calculus
First, give them an example of "trying to do it without calculus". It'd look like this:
The acceleration of a particle at time $t$ seconds is, say, $a=t^2$.
What's the change in velocity ...
- 6,427
20
votes
Why use Fourier series instead of Taylor?
Great question. One reason that complex exponential expansions (which end up turning on sines and cosines for real-valued problems) are more natural that Taylor series expansions is that they don't ...
- 48.8k
20
votes
Accepted
The usage of chain rule in physics
You are correct that you cannot (globally) write velocity as a function of distance. For example, as one commenter has already mentioned, throw a ball directly up in the air and wait for it to come ...
- 16.3k
20
votes
Why do we need instantaneous speed?
Instantaneous speed is what the police officer fines you for. On a trip to the bakery it doesn't really matter that your average speed is 40 km/hr if you during the trip reached 120 km/hr for just a ...
- 51.7k
18
votes
Accepted
Why does acceleration need to be constant if integrating?
Acceleration does not need to be constant. By definition, $a=dv/dt$. You can still solve for $v(t)$ by integrating $\int a(t) dt$.
If acceleration is constant, you will arrive at the common ...
- 1,687
18
votes
Accepted
Can acceleration depend linearly on velocity?
Yes, air resistance at low velocities is one such example, but I'm sure that there are others. For an object moving at a suitably low velocity, the drag on the object is given by $$\vec{F} = - b \vec{...
- 917
17
votes
Explaining how we cannot account for changing acceleration questions without calculus
On a historical note, Isaac Newton studied the same problem for gravity, and presented the solution in Principia Mathematica without using calculus, although it turns out that he had privately ...
- 171
17
votes
Can $d/t =$ speed ever be wrong? Is there a more accurate way to determine speed?
I'm very sorry for your loss.
The definition of the average speed of an object as it passes between two points is the distance $d$ between them divided by the time $t$ it took to get from one to the ...
- 70.5k
16
votes
How area under Velocity-Time graph represents magnitude of displacement?
Imagine dividing your graph of velocity vs. time into a bunch of extremely thin vertical rectangles. It's reasonable to say that, over such a short time, velocity is constant in any given rectangle. ...
- 35.7k
16
votes
Question about derivation of kinematics equations
Given velocity $v(t)$, the distance moved after a certain time $t$ is not $v(t)t$ - this formula works at constant velocity, but when the velocity is changing, the correct expression is $\int^{t_f}_{...
- 21.6k
16
votes
How is it possible to differentiate or integrate with respect to discrete time or space?
This is a comment, as Andrew's answer is adequate for the problem.
I want to point out , which is not clear in your question, the difference between mathematical modeling and the object modeled.
When ...
- 235k
16
votes
What is the instant velocity?
Average speed is defined as passed-distance-over-passed-time:
$$v_\text{average}=\frac{\Delta s}{\Delta t}.$$
In other words, choose a point on your path. Then choose one more point. Plug in the ...
- 51.7k
16
votes
In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
The notation is a little sloppy from a purely mathematical point of view (although common in physics) so it might be causing a little confusion.
To help clarify, it might help to use different letters ...
- 52.3k
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