SHOW WORK PLEASE!!!
The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 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) 1,2
ii) 1,1.1
iii)1,1.01
iv)1, 1.001
To find the average velocity during each time period, we need to calculate the displacement during that time period and divide it by the duration of the time period.
(a)
i) Time period: 1 to 2 seconds
Displacement between t = 1 and t = 2:
s(2) - s(1) = 2sin(π(2)) + 3cos(π(2)) - [2sin(π(1)) + 3cos(π(1))]
First, let's calculate the values at t = 1 and t = 2:
s(1) = 2sin(π(1)) + 3cos(π(1))
s(2) = 2sin(π(2)) + 3cos(π(2))
Using a calculator, we find:
s(1) ≈ 2.00 cm
s(2) ≈ 0.00 cm
Now we can calculate the displacement:
s(2) - s(1) ≈ 0.00 cm - 2.00 cm
≈ -2.00 cm
The duration of the time period is 2 - 1 = 1 second.
Average velocity = Displacement / Time duration
Average velocity = -2.00 cm / 1 second
Average velocity ≈ -2.00 cm/s
ii) Time period: 1 to 1.1 seconds
Displacement between t = 1 and t = 1.1:
s(1.1) - s(1) = 2sin(π(1.1)) + 3cos(π(1.1)) - [2sin(π(1)) + 3cos(π(1))]
Calculating the values at t = 1 and t = 1.1:
s(1) ≈ 2.00 cm
s(1.1) ≈ 2sin(π(1.1)) + 3cos(π(1.1))
Using a calculator, we find:
s(1.1) ≈ 1.95 cm
Displacement:
s(1.1) - s(1) ≈ 1.95 cm - 2.00 cm
≈ -0.05 cm
The duration of the time period is 1.1 - 1 = 0.1 second.
Average velocity = Displacement / Time duration
Average velocity = -0.05 cm / 0.1 second
Average velocity ≈ -0.50 cm/s
iii) Time period: 1 to 1.01 seconds
Displacement between t = 1 and t = 1.01:
s(1.01) - s(1) = 2sin(π(1.01)) + 3cos(π(1.01)) - [2sin(π(1)) + 3cos(π(1))]
Calculating the values at t = 1 and t = 1.01:
s(1) ≈ 2.00 cm
s(1.01) ≈ 2sin(π(1.01)) + 3cos(π(1.01))
Using a calculator, we find:
s(1.01) ≈ 1.999 cm
Displacement:
s(1.01) - s(1) ≈ 1.999 cm - 2.00 cm
≈ -0.001 cm
The duration of the time period is 1.01 - 1 = 0.01 second.
Average velocity = Displacement / Time duration
Average velocity = -0.001 cm / 0.01 second
Average velocity ≈ -0.10 cm/s
iv) Time period: 1 to 1.001 seconds
Displacement between t = 1 and t = 1.001:
s(1.001) - s(1) = 2sin(π(1.001)) + 3cos(π(1.001)) - [2sin(π(1)) + 3cos(π(1))]
Calculating the values at t = 1 and t = 1.001:
s(1) ≈ 2.00 cm
s(1.001) ≈ 2sin(π(1.001)) + 3cos(π(1.001))
Using a calculator, we find:
s(1.001) ≈ 1.9999 cm
Displacement:
s(1.001) - s(1) ≈ 1.9999 cm - 2.00 cm
≈ -0.0001 cm
The duration of the time period is 1.001 - 1 = 0.001 second.
Average velocity = Displacement / Time duration
Average velocity = -0.0001 cm / 0.001 second
Average velocity ≈ -0.10 cm/s
a,b,c,d put all the value in the formula than divide the time :
a) figure out s(2) and s(1) than divide
s(2) = 3sin(pi*2) + 4cos(pi2) = 3(0) +4(1) = 4
s(1) = 3sin(pi1) +4cos(pi1) = 3(0) +4(-1) = -4
average velocity is ( s2-s1)/(2-1) = (4- (-4))/ 1 = 8 cm/s
b) {s(1.1) -s(1)}/(1.1 -1) = -7.3 cm/s
c) { s( 1.01)-s(1)}/(1.01-1) = -9.2 cm/ s
d) {s(1.004)-s(1)}/(1.004-1) = -9.3
e) take derivative then put t =1
3picos(pit)-4pisin(pit)
3picos(pi1)-4pisin(pi1)
3pi(-1) = -3pi = -9.42 cm/s
you can estimate t= 1 when they go small interval 1.0005 and so on they will approach to -9.4m/s bylooking
note part b,c,d are the same part a, i just give you an answer but you have to work out like part a to show your works. you just punching in your calculator to figure out your answer by yourself
s = 4 sin(πt) + 3 cos(πt)
as you know, the average velocity is the integral divided by the time interval. Since the position is the integral of the velocity, the average velocity in [a,b] = (s(b)-s(a))/(b-a)
So, we have
i) (s(2)-s(1))/(2-1) = (2sin2π+3)-(2sinπ+3) = 0
ii) (s(1.1)-s(1))/(1.1-1) = ((2sin1.1π+3)-(2sin.1π+3)))/.1 = -6.18034
and so on