   Chapter 2, Problem 37RE

Chapter
Section
Textbook Problem

The displacement (in meters) of an object moving in a straight line is given by s = 1 + 2t + ¼ t2, where t is measured in seconds.(a) Find the average velocity over each time period.(i) [l, 3](ii) [1, 2](iii) [ 1, 1.5](iv) [ 1, 1.1](b) Find the instantaneous velocity when t = 1.

(a)

To determine

To find: The average velocity of an object over the given time periods.

Explanation

Given:

The displacement (in meters) of an object is moving in a straight line.

The equation of a position function is s=1+2t+14t2, where t is measured in seconds.

Formula used:

The average velocity over the time interval [a,a+h] is,

f(a+h)f(a)h (1)

Calculation:

Obtain the average velocity of the particle over the time interval [a,a+h].

Take the position function s(t)=f(t) and use equation (1) to compute the average velocity over the time interval [a,a+h] as follows.

Average velocity=f(a+h)f(a)h=(1+2(a+h)+14(a+h)2)(1+2(a)+14(a)2)h=1+2a+2h+14(a+h)212(a)14(a)2h=2h+14(a+h)214a2h

Simplify further and obtain the value of average velocity as follows.

Average velocity=2h+14(a2+h2+2ah)14a2h=2h+14a2+14h2+12ah14a2h=2h+14h2+12ahh=h(2+14h+12a)h

=2+14h+12a

Thus, the average velocity of the particle over the time interval [a,a+h] is (2+14h+12a) m/s_.

Section (i)

Obtain the average velocity of the particle over the time interval [1,3].

Consider a=1 and a+h=3.

h=3a=3(1)=2

Substitute 1 for a and 2 for h in the average velocity,

Average velocity=2+14h+12a=2+14(2)+12(1)=2+12+12=3

Thus, the average velocity of the particle over the time interval [1,3] is 3 m/s

(b)

To determine

To find: The instantaneous velocity at t=1 second.

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