   Chapter 13, Problem 17RE

Chapter
Section
Textbook Problem

# A particle moves with position function r(t) = t ln t i + t j + e−t k. Find the velocity, speed, and acceleration of the particle.

To determine

To find: The velocity of a particle that moves with the position function r(t)=tlnti+tj+etk , speed of a particle that moves with the position function r(t)=tlnti+tj+etk and acceleration of a particle that moves with the position function r(t)=tlnti+tj+etk .

Explanation

Given data:

The particle moves with the position function r(t)=tlnti+tj+etk .

Formula used:

Write the expression to find the velocity with the position function r(t) :

v(t)=ddt[r(t)] (1)

Write the required differentiation formulae to obtain the solution as follows.

ddt[u(t)v(t)]=u(t)ddt[v(t)]+v(t)ddt[u(t)]ddt(lnt)=1tddt(tn)=ntn1ddt(et)=et

Write the expression to find the speed of a particle with the position function r(t) .

speed=|v(t)| (2)

Here,

v(t) is the velocity of the particle.

Write the expression to find the acceleration of a particle with the position function r(t) .

a(t)=ddt[v(t)] (3)

Substitute (tlnti+tj+etk) for r(t) in equation (1),

v(t)=ddt(tlnti+tj+etk)=ddt(tlnt)i+ddt(t)j+ddt(et)k=[tddt(lnt)+lntddt(t)]i+(1)j+(et)k=[t(1t)+lnt(1)]i+j(et)k

Simplify the expression as follows.

v(t)=(1+lnt)i+jetk

Thus, the velocity of a particle that moves with the position function r(t)=tlnti+tj+etk is (1+lnt)i+jetk_

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