Answered: The roller coaster car travels down the…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

The roller coaster car travels down the helical path at constant speed such that the parametric equiations that define its position are

x = c sin kt , y = c cos kt, z= h - bt, where c, h and b are constants.

Determine the magnitudes of its velocity and acceleration.

Expert Solution

Step 1

Given data as pr question

x = c sin kt ,

y = c cos kt,

z= h - bt,

where c, h, and b are constants.

Component of the velocity of the roller

$\begin{array}{l} v_{x} = \frac{d x}{d t} \\ = \frac{d (c \sin k t)}{d t} \\ = c k . \cos (k t) \\ simillarly for v_{y} and v_{z}, w e g e t \\ v_{y} = - c k . \sin (k t) \\ v_{z} = - b \\ N o w \\ a_{x} = \frac{d v}{d t} \\ = \frac{d (c k . \cos k t)}{d t} \\ = - c k^{2} . \sin (k t) \\ simillarly for a_{y} and a_{z}, w e g e t \\ a_{y} = - c k^{2} . \cos (k t) \\ a_{z} = 0 \end{array}$

$\begin{array}{l} w r i t t i n g e q u a t i o n s f o r v e l o c i t y a n d a c c e l e r a t i o n \\ v_{x} = c k . \cos (k t) \\ a_{x} = - c k^{2} . \sin (k t) \\ v_{y} = - c k . \sin (k t) \\ a_{y} = - c k^{2} . \cos (k t) \\ v_{z} = - b \\ a_{z} = 0 \end{array}$

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