The displacement (in centimeters) of a particle s moving back and forth along a straight line is given by the equation s = 3 sin(πt) + 3 cos(πt), where t is measured in seconds. (Round your answers to two decimal places.)
(a)Find the average velocity during each time period.
(i)[1, 1.01]cm/s
(ii)[1, 1.001]cm/s
(b)Estimate the instantaneous velocity of the particle when t = 1.
when t = 1, s = 3sinΟ + 3 cosΟ = 3(0) + 3(-1) = -3
when t = 1.01, s = 3sin(1.01Ο) + 3cos(1.01Ο) = -3.0927
when t = 1.001, s = 3sin(1.001Ο) + 3cos(1.001Ο) = -3.00949958
avg velocity between t = 1 and t = 1.01 = (-3.0927 + 3)/(1.01-1)
= appr -9.28
avg velocity between t = 1 and t = 1.001 = (-3.0094... + 3)/.001
= appr -9.41
btw, just in case you may be interested -3Ο = -9.425
mmmhhh?
(a) To find the average velocity during each time period, we need to find the displacement of the particle at the beginning and end of the time period and divide it by the length of the time period.
(i) [1, 1.01]cm/s
First, let's calculate the displacement at t = 1 and t = 1.01:
s(1) = 3sin(π*1) + 3cos(π*1) = 3sin(π) + 3cos(π) = 3(0) + 3(-1) = -3 cm
s(1.01) = 3sin(π*1.01) + 3cos(π*1.01) = 3sin(1.01π) + 3cos(1.01π) β -2.97 cm
The displacement during this time period is s(1.01) - s(1) β -2.97 - (-3) = 0.03 cm
Next, let's calculate the length of the time period:
Time period = 1.01 - 1 = 0.01 s
Finally, let's calculate the average velocity:
Average velocity = displacement / time period β 0.03 cm / 0.01 s β 3 cm/s
(ii) [1, 1.001]cm/s
Using the same calculation method, we find that the displacement during this time period is approximately 0.003 cm and the time period is 0.001 s. Therefore, the average velocity is:
Average velocity β displacement / time period β 0.003 cm / 0.001 s β 3 cm/s
(b) To estimate the instantaneous velocity of the particle when t = 1, we need to calculate the derivative of the displacement equation with respect to time and substitute t = 1 into it.
The derivative of s with respect to t is:
s'(t) = 3πcos(πt) - 3πsin(πt)
Substituting t = 1 into the derivative equation:
s'(1) = 3πcos(π*1) - 3πsin(π*1) = 3πcos(π) - 3πsin(π) = 3π(0) - 3π(1) = -3π
Therefore, the estimated instantaneous velocity of the particle when t = 1 is approximately -3π cm/s.
To find the average velocity during each time period, we need to calculate the change in displacement and divide it by the change in time.
(a)
(i) [1, 1.01]
To find the change in displacement, we need to evaluate the equation at the endpoints of the time interval:
s(1) = 3sin(π(1)) + 3cos(π(1)) = 3sin(π) + 3cos(π) = 3(0) + 3(-1) = -3 cm
s(1.01) = 3sin(π(1.01)) + 3cos(π(1.01)) β 3sin(3.14) + 3cos(3.14) β 3(0.0016) + 3(-0.9999) β -2.9983 cm
The change in displacement is: (-2.9983) - (-3) = 0.0017 cm
The change in time is: 1.01 - 1 = 0.01 s
Average velocity = change in displacement / change in time = 0.0017 cm / 0.01 s β 0.17 cm/s
(ii) [1, 1.001]
To find the change in displacement, we need to evaluate the equation at the endpoints of the time interval:
s(1) = -3 cm
s(1.001) β -3 cm
The change in displacement is: (-3) - (-3) = 0 cm
The change in time is: 1.001 - 1 = 0.001 s
Average velocity = change in displacement / change in time = 0 cm / 0.001 s = 0 cm/s
(b)
To estimate the instantaneous velocity of the particle when t = 1, we can take the derivative of the displacement equation with respect to time:
v(t) = d/dt [3sin(πt) + 3cos(πt)]
= 3πcos(πt) - 3πsin(πt)
Substituting t = 1:
v(1) = 3πcos(π(1)) - 3πsin(π(1))
= 3πcos(π) - 3πsin(π)
= 3π(-1) - 3π(0)
= -6π
Approximating π as 3.14, the instantaneous velocity when t = 1 is approximately -6(3.14) β -18.85 cm/s.
To find the average velocity during a time period, you need to determine the displacement of the particle during that time period and divide it by the duration of the time period.
(a) Let's find the average velocity during the time period [1, 1.01].
1. Calculate the displacement of the particle at the start and end of the time period:
- At t = 1: s = 3 sin(π(1)) + 3 cos(π(1)) = 3(0) + 3(-1) = -3 cm
- At t = 1.01: s = 3 sin(π(1.01)) + 3 cos(π(1.01)) = 3(0.0314) + 3(-0.0314) = -0.028 cm
2. Calculate the duration of the time period:
The duration = 1.01 - 1 = 0.01 s
3. Calculate the average velocity:
Average velocity = (Displacement) / (Duration) = (-0.028 - (-3)) / 0.01 cm/s
The average velocity during the time period [1, 1.01]cm/s is approximately 297.20 cm/s (rounded to two decimal places).
Let's now find the average velocity during the time period [1, 1.001].
1. Calculate the displacement of the particle at the start and end of the time period:
- At t = 1: s = -3 cm (same as before)
- At t = 1.001: s = 3 sin(π(1.001)) + 3 cos(π(1.001)) = 3(0.0031) + 3(-0.0031) = -0.00277 cm
2. Calculate the duration of the time period:
The duration = 1.001 - 1 = 0.001 s
3. Calculate the average velocity:
Average velocity = (Displacement) / (Duration) = (-0.00277 - (-3)) / 0.001 cm/s
The average velocity during the time period [1, 1.001]cm/s is approximately 2997.23 cm/s (rounded to two decimal places).
(b) To estimate the instantaneous velocity of the particle when t = 1, we can calculate the derivative of the displacement equation with respect to time (t).
1. Differentiate the equation: ds/dt = 3π cos(πt) - 3π sin(πt)
2. Substitute t = 1 into the derivative equation:
ds/dt = 3π cos(π(1)) - 3π sin(π(1))
Simplifying further,
ds/dt = 3π cos(π) - 3π sin(π)
Applying trigonometric values,
ds/dt = 3π(0) - 3π(1)
= -3π
The instantaneous velocity of the particle when t = 1 is estimated to be -3π cm/s.