The navigation rules for countries in the Unites States are compiled in the U.S. Coast Guard. There are actually boating courses that are recognized by the U.S Coast Guard. One of the things they discuss there is the emergency aids every boat should have depending on the size and purpose. For recreational vessels with lengths measuring less than 39.4 feet, they must carry aboard an efficient sound producing device. This will serve as your signal when emergencies happen and you have to get the attention of nearby boat vessels. Hence, for practical purposes, every boat should have a whistle or a horn.

4 0

2 years ago

A particle’s position is ~r = (ct2 − 2dt3 )iˆ+ (2ct2 − dt2 )jˆ, where c and d are positive constants. Find the expressions for t

Mars2501 [29]

The velocity of the particle is given by the derivative of the position vector:

$\vec v = \dfrac{\mathrm d\vec r}{\mathrm dt} = (2ct-6dt^2)\,\vec\imath + (4ct-2dt)\,\vec\jmath$

(a) The particle is moving in the x-direction when the y-component of velocity is zero:

$4ct-2dt = 2t (2c - d) = 0 \implies t=0$

But we want t > 0, so this never happens, unless 2c = d is given, in which case the y-component is always zero.

(b) Similarly, the particle moves in the y-direction when the x-component vanishes:

$2ct-6dt^2 = 2t (c - 3dt) = 0 \implies t=0 \text{ or }c-3dt = 0$

We drop the zero solution, and we're left with

$c-3dt = 0 \implies c=3dt \implies \boxed{t = \dfrac c{3d}}$

In the case of 2c = d, this times reduces to t = c/(6c) = 1/6.

7 0

2 years ago