Physics, 01.10.2019 11:20 amortegaa805

Include instructor prompts. what does your instructor want you to accomplish? a stone is thrown from the top of a slope that makes an angle of 35 degrees with the horizontal. at what angle to the horizontal should the stone be thrown to get a maximum range?

Answers: 1

Show answers

Answers

[see the attached figure for a better understanding of my notations]

All the angles below are in radians unless written otherwise

: mass of the ball
$\alpha$ : angle by which the ground is inclined (here $a=-35^\circ$ )
$\beta$ : initial angle of the throw

: acceleration due to gravity, $g\approx9.81m^2/s$
$\vec{y}$ : vertical unit vector, pointing upwards
$\vec{x}$ : horizontal unit vector, pointing to the right
x,y

: x,y-position of the ball
$\vec{a}$ : ball's acceleration
$\vec{v}$ : ball's speed
$\vec{p}$ : ball's position
$\vec{v_0}$ : initial ball's speed

I don't now how much you know in physics, so I'll try to break this down as much as possible.

The only force that is applied on the ball when it's falling is gravity $\vec{P}$ . By Newton's second law, $m\vec{a}=\vec{P}=-mg\vec{y}$ hence by integrating with respect to time, $\vec{v}=-gt\vec{y}+\vec{v_0}$ which by integrating once more yields $\vec{p}=-g\frac{t^2}2\vec{y}+\vec{v_0}t$

Thus $y=-g\frac{t^2}2+v_0\sin(\beta)t,x=v_0\cos(\beta)t$ . That last one gives $t=\dfrac{x}{v_0\cos(\beta)}$ hence $y=-g\dfrac{x^2}{2v_0^2\cos^2(\beta)}+x\tan(\beta)$

The ball stops when it intersects with the slope defined by $y=\tan(\alpha)x$ hence if we call

the abscissa of the the intersection point, $\tan(\alpha)X=-g\dfrac{X^2}{2v_0^2\cos^2(\beta)}+X\tan(\beta)$ hence $X(\beta)=\dfrac{(b-a)2v_0^2\cos^2(\beta)}{g}$ where $a=\tan(\alpha),b=\tan(\beta)$ .

To find the maximum of $X(\beta)$ , we derive the function and solve $\dfrac{dX}{d\beta}=0$ which is equivalent to $0=\dfrac{d((\tan(\beta)-a)\cos^2(\beta))}{d\beta}=1-(b-a)\sin(2\beta)$ . Using the well known formula $\sin(2\beta)=\dfrac{2b}{1+b^2}$ we get b^2-2ab-1=0

. The discriminant of this equation is $\Delta=4(a^2+1)$ hence the unique positive solution is $b=a+\sqrt{a^2+1}$ thus $\beta=\arctan(a+\sqrt{a^2+1})$

In our case $a=\tan(\dfrac{-35*2\pi}{360})=-\tan(\dfrac{7\pi}{36})$ hence $b=\arctan\left(a+\sqrt{a^2+1}\right)=27.5^\circ$ , which is the optimum angle to get a maximum range.
Include instructor prompts. what does your instructor want you to accomplish? a stone is thrown fro

Include instructor prompts. what does your instructor want you to accomplish? a stone is thrown fro

Answer from: Quest

relativism, roughly put, is the view that truth and falsity, right and wrong, standards of reasoning, and procedures of justification are products of differing conventions and frameworks of assessment and that their authority is confined to the context giving rise to them. more precisely, “relativism” covers views which maintain that—at a high level of abstraction—at least some class of things have the properties they have (e.g., beautiful, morally good, epistemically justified) not simpliciter, but only relative to a given framework of assessment (e.g., local cultural norms, individual standards), and correspondingly, that the truth of claims attributing these properties holds only once the relevant framework of assessment is specified or supplied. relativists characteristically insist, furthermore, that if something is only relatively so, then there can be no framework-independent vantage point from which the matter of whether the thing in question is so can be established.

relativism has been, in its various guises, both one of the most popular and most reviled philosophical doctrines of our time. defenders see it as a harbinger of tolerance and the only ethical and epistemic stance worthy of the open-minded and tolerant. detractors dismiss it for its alleged incoherence and uncritical intellectual permissiveness. debates about relativism permeate the whole spectrum of philosophical sub-disciplines. from ethics to epistemology, science to religion, political theory to ontology, theories of meaning and even logic, philosophy has felt the need to respond to this heady and seemingly subversive idea. discussions of relativism often also invoke considerations relevant to the very nature and methodology of philosophy and to the division between the so-called “analytic and continental” camps in philosophy. and yet, despite a long history of debate going back to plato and an increasingly large body of writing, it is still difficult to come to an agreed definition of what, at its core, relativism is, and what philosophical import it has. this entry attempts to provide a broad account of the many ways in which “relativism” has been defined, explained, defended and criticized.

explanation:

Answer from: Quest

the minimum amperage that the current in this wire would have to have so that its weight is completely supported by the magnetic force due to earth's field, assuming that no other forces except gravity act on it