Recent content by kuruman

The calculator assumes that you have a rotating floor such that the angular velocity of rotation is perpendicular to the floor. It provides a path given an initial velocity and position for a mass moving without friction across the floor. It is not the hollow asteroid that you mentioned in post...

Can you be a bit more specific about the fountain environment? I suppose that the asteroid can be modeled as a spherical shell in which the gravitational field is zero. You put a fountain inside as shown in post #30. Let ##\vec L## be a vector of magnitude 110 m pointing from the base of the...

So when OP writes what does that mean? To me it says that OP rejects plotting ##v^2## vs. ##x## and is seeking an alternate way.

Isn't this a circular argument? How does one get this expression to begin with? The work-energy theorem is a rewritten "third equation" once Newton's second law is established and comes after kinematics in the standard curriculum. The point is one of idle curiosity. The question is not to do...

That's my point which, I assume, is also OP's query. Is there a way to know directly and plot ##v^2(x)## independently of knowing ##v(t)## and/or doing arithmetic that relies on the previous two equations?

The opening statement of the problem declares "The three equations describing one-dimensional kinematics are well known from high school." Based on my high school experience, these equations are usually given as "that's the way it is" instead of solving the ODE ##~\ddot x = \text{const.}~##...

I think that this approach requires collection of experimental values for ##v## and ##x##, however . . . Can a fit to the data be considered a "derivation"?

As @Ibix remarked, some algebra is inevitable when you translate a graphic form into an algebraic form. Here is the simplest way I can think of to do this using a velocity vs. time plot (see figure on the right). Total distance ##x## traveled in time ##t## is "the area under the curve", in...

Sure, I agree that the drawing has mislabeled the positive and negative arcs on the ring, but how does that affect the integrals' setup? OP has not posted the rest of the solution, but I suspect that the (mis)labeling of the arcs is not part of it.

First of all, the charge density is given as ##\lambda(\phi)=\lambda_0\cos(\phi)## where ##\phi## is the azimuthal angle as shown in the diagram. Quantity ##x## is the distance from the center of the ring to the point of interest along the ##x##-axis. That kind of carelessness is not expected...

You need to be more more specific about what you don't understand otherwise we cannot help you. What is vague about the diagram? It shows how the electric field vector element ##d\mathbf E## is resolved along the three Cartesian axes ##x##, ##y## and ##z##. What charge density have they not...

I always thought that the speed of sound in air is about 340 m/s and used the rule "three seconds per kilometer" for a quick and dirty calculation.

Looks good.

Euler's equations for rigid body dynamics allow torque-free precession. An often-seen problem in intermediate Classical Mechanics is to show that when a rigid body with three different principal moments of inertia is started spinning along each one of the axes in free space, the motion will be...

If it isn't simpler, it certainly is more elegant.