The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motions

= 2 sin(π t) + 2 cos(π t),

where t is measured in seconds. (Round your answers to two decimal places.)

Find the average velocity during each time period.

(i) [1, 2] (1 to 2 seconds)

____cm/s

(ii) [1, 1.1]

____ cm/s

(iii) [1, 1.01]

____cm/s

(iv) [1, 1.001]

____cm/s

(b) Estimate the instantaneous velocity of the particle when t = 1.

____cm/s

Ruublrr Jan 29, 2019

#1**+1 **

2[ sin (pi *t ) + cos (pi * t)

From 1 to 2 seconds we have

[ 2 sin (2pi) + 2cos (2 pi) - 2sin (pi) - 2cos(pi)] / (2 - 1) =

[ 2 - (-2) ] / 1 = 4 cm /sec

From 1 to 1.1 seconds

[ 2 sin (1.1pi) + 2cos (1.1 pi) - 2sin (pi) - 2cos(pi)] / (1.1 - 1) ≈ -5.20 cm/sec

From 1 to 1.01 seconds

[ 2 sin (1.01*pi) + 2cos (1.01* pi) - 2sin (pi) - 2cos(pi)] / (1.01 - 1) ≈ -6.18 cm/sec

From 1 to 1.001 seconds

[ 2 sin (1.001*pi) + 2cos (1.001* pi) - 2sin (pi) - 2cos(pi)] / (1.001 - 1) ≈ - 6.27 cm /sec

It appears that the instantaneous velocity at 1 sec = -2pi cm/sec ≈ -6.28 cm/sec

BTW....we can verify that this is true using some Calculus !!!

CPhill Jan 29, 2019