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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the velocity, speed, and acceleration of a particle moving with position function r(t) = (2t2 − 3) i + 2t j. Sketch the path of the particle and draw the position, velocity, and acceleration vectors for t = 1.

To determine

To find: The velocity of a particle that moves with the position function r(t)=(2t23)i+2tj , speed of a particle that moves with the position function r(t)=(2t23)i+2tj , acceleration of a particle that moves with the position function r(t)=(2t23)i+2tj and path of the particle, position, velocity and acceleration vectors of the particle for t=1 .

Explanation

Given data:

The particle moves with the position function r(t)=(2t23)i+2tj .

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)]=ddt[u(t)]+ddt[v(t)]ddttn=ntn1

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

speed=|v(t)| (2)

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

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

Here,

v(t) is the velocity of the particle.

Substitute (2t23)i+2tj for r(t) in equation (1),

v(t)=ddt[(2t23)i+2tj]=ddt(2t23)i+ddt(2t)j=[ddt(2t2)+ddt(3)]i+(2)j=[2(2t)+0]i+(2)j

Simplify the expression as follows.

v(t)=4ti+2j

Thus, the velocity of a particle that moves with the position function r(t)=(2t23)i+2tj is 4ti+2j_ .

Calculation of speed of a particle:

Substitute (4ti+2j) for v(t) in equation (2),

speed=|(4ti+2j)|=(4t)2+22=16t2+4=4(4t2+1)

speed=24t2+1

Thus, the speed of a particle that moves with the position function r(t)=(2t23)i+2tj is 24t2+1_

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