Show the work for a, b, and c

Question

The displacement (in centimeters) of a particle moving back and forth along a line is given by the equation of motion 𝑠(𝑡)=3sin(𝜋𝑡)+2cos(𝜋𝑡), where 𝑡is measured in seconds.

(a)Write a formula for the average velocity of the ball for 𝑡near 1.

(b)Find the average velocity during each time period: [1,2];[1,2.001];[1,1.1];[1,1.01];[1,1.001]. Give each value rounded to 3decimals and with correct units.

(c)Estimate for the instantaneous velocity of the particle at 𝑡=1𝑠?

Answer 1

This is how I would approach this problem.

a) The formula for the average velocity is the slope of the secant line. Since we are asked to find the formula near 1, I would write:

a(t) = [s(t)-s(1)]/(t-1)

Since s(1)= 3sin(π)+2cos(π) = 0-2=-2

We could write a(t) = [s(t)+2]/(t-1)

b) Use the formula from part a:

on [1, 2]: [s(2)+2]/(2-1) = [3sin(2π)+2cos(2π)+2]/(1) = 4 cm/s

on [1, 2.001]: [s(2.001)+2]/(2.001-1) = [3sin(2.001π)+2cos(2.001π)+2]/(1.001) = 4.005 cm/s

on [1, 1.1]: [s(1.1)+2]/(1.1-1) = [3sin(1.1π)+2cos(1.1π)+2]/(0.1) = -8.292 cm/s

on [1, 1.01]: [s(1.01)+2]/(1.01-1) = [3sin(1.01π)+2cos(1.01π)+2]/(0.01) = -9.325 cm/s

on [1, 1.001]: [s(1.001)+2]/(1.001-1) = [3sin(1.001π)+2cos(1.001π)+2]/(0.001) = -9.415 cm/s

c) v(t) = ds/dt = 3πcos(πt)-2πsin(πt)

v(1)=3πcos(π)-2πsin(π)= -3π cm/s ≈ -9.425 cm/s

Notice how the answers for (b) are approaching the answer to (c).

I hope this helps!