The position coordinates of a particle moving | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The velocity components are obtained by differentiating the position coordinates with respect to time $t$:$v_x = \frac{dx}{dt} = -a \omega \sin \o

Vedclass · Accepted Answer

$\sqrt{2} a \omega$

Vedclass · Answer

(A) The velocity components are obtained by differentiating the position coordinates with respect to time $t$:$v_x = \frac{dx}{dt} = -a \omega \sin \omega t$$v_y = \frac{dy}{dt} = a \omega \cos \omega t$$v_z = \frac{dz}{dt} = a \omega$The speed $v$ is the magnitude of the velocity vector:$v = \sqrt{v_x^2 + v_y^2 + v_z^2}$$v = \sqrt{(-a \omega \sin \omega t)^2 + (a \omega \cos \omega t)^2 + (a \omega)^2}$$v = \sqrt{a^2 \omega^2 \sin^2 \omega t + a^2 \omega^2 \cos^2 \omega t + a^2 \omega^2}$$v = \sqrt{a^2 \omega^2 (\sin^2 \omega t + \cos^2 \omega t) + a^2 \omega^2}$Since $\sin^2 \omega t + \cos^2 \omega t = 1$,we get:$v = \sqrt{a^2 \omega^2 (1) + a^2 \omega^2} = \sqrt{2 a^2 \omega^2} = \sqrt{2} a \omega$

The position coordinates of a particle moving in a $3-D$ coordinate system are given by $x = a \cos \omega t$,$y = a \sin \omega t$,and $z = a \omega t$. The speed of the particle is:

Similar Questions

$A$ body starts from rest from the origin with an acceleration of $6 \; m/s^2$ along the $x$-axis and $8 \; m/s^2$ along the $y$-axis. Its distance from the origin after $4 \; s$ will be (in $; m$)

What can be the angle between velocity and acceleration for the motion in two or three dimensions?

The magnitude of acceleration and velocity of a particle moving in a plane,whose position vector is $\vec{r} = 3t^2 \hat{i} + 2t \hat{j} + \hat{k}$ at $t = 2 \text{ s}$,are,respectively:

The position vector of a moving body at any instant of time is given as $\overrightarrow{r} = (5t^2 \hat{i} - 5t \hat{j}) \text{ m}$. The magnitude and direction of velocity at $t = 2 \text{ s}$ is,

Vedclass Test Series

Exam Paper Generator

Online Exam Module