The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion
s = 3 sin(πt) + 5 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 change in displacement divided by the change in time within the given time interval.
(a) For the time interval [1, 2]:
(i) [1, 2]
To calculate the average velocity, we need to find the change in displacement and time.
The initial time, t1 = 1 sec, and the final time, t2 = 2 sec.
To find the change in displacement, we need to find s2 - s1, where s2 is the displacement at time t=2, and s1 is the displacement at time t=1.
s = 3 sin(πt) + 5 cos(πt)
Let's substitute the initial and final times into the equation:
s1 = 3 sin(π * 1) + 5 cos(π * 1)
s1 = 3 sin(π) + 5 cos(π)
s1 = 0 + 5 (-1)
s1 = -5
s2 = 3 sin(π * 2) + 5 cos(π * 2)
s2 = 3 sin(2π) + 5 cos(2π)
s2 = 0 + 5 (1)
s2 = 5
Now we can calculate the change in displacement:
Δs = s2 - s1
Δs = 5 - (-5)
Δs = 10 cm
The change in time is Δt = t2 - t1 = 2 - 1 = 1 sec
Now let's calculate the average velocity:
Average velocity = Δs / Δt
Average velocity = 10 cm / 1 sec
Average velocity = 10 cm/sec
Therefore, the average velocity during the time period [1, 2] is 10 cm/sec.
(ii) [1, 1.1]
For this time interval, we follow the same steps as in part (i).
The initial time, t1 = 1 sec, and the final time, t2 = 1.1 sec.
Substitute the values into the equation for displacement:
s1 = 3 sin(π * 1) + 5 cos(π * 1)
s1 = 0 + 5 (-1)
s1 = -5
s2 = 3 sin(π * 1.1) + 5 cos(π * 1.1)
s2 ≈ 4.37
Now we calculate the change in displacement:
Δs = s2 - s1
Δs ≈ 4.37 - (-5)
Δs ≈ 9.37 cm
The change in time is Δt = t2 - t1 = 1.1 - 1 = 0.1 sec
Now let's calculate the average velocity:
Average velocity = Δs / Δt
Average velocity = 9.37 cm / 0.1 sec
Average velocity ≈ 93.7 cm/sec
Therefore, the average velocity during the time period [1, 1.1] is approximately 93.7 cm/sec.
(iii) [1, 1.01]
For this time interval, follow the same steps as in part (i).
The initial time, t1 = 1 sec, and the final time, t2 = 1.01 sec.
Substitute the values into the equation for displacement:
s1 = 3 sin(π * 1) + 5 cos(π * 1)
s1 = 0 + 5 (-1)
s1 = -5
s2 = 3 sin(π * 1.01) + 5 cos(π * 1.01)
s2 ≈ -4.93
Now we calculate the change in displacement:
Δs = s2 - s1
Δs ≈ -4.93 - (-5)
Δs ≈ 0.07 cm
The change in time is Δt = t2 - t1 = 1.01 - 1 = 0.01 sec
Now let's calculate the average velocity:
Average velocity = Δs / Δt
Average velocity = 0.07 cm / 0.01 sec
Average velocity = 7 cm/sec
Therefore, the average velocity during the time period [1, 1.01] is 7 cm/sec.
(iv) [1, 1.001]
For this time interval, follow the same steps as in part (i).
The initial time, t1 = 1 sec, and the final time, t2 = 1.001 sec.
Substitute the values into the equation for displacement:
s1 = 3 sin(π * 1) + 5 cos(π * 1)
s1 = 0 + 5 (-1)
s1 = -5
s2 = 3 sin(π * 1.001) + 5 cos(π * 1.001)
s2 ≈ -4.99987
Now we calculate the change in displacement:
Δs = s2 - s1
Δs ≈ -4.99987 - (-5)
Δs ≈ 0.00013 cm
The change in time is Δt = t2 - t1 = 1.001 - 1 = 0.001 sec
Now let's calculate the average velocity:
Average velocity = Δs / Δt
Average velocity = 0.00013 cm / 0.001 sec
Average velocity = 0.13 cm/sec
Therefore, the average velocity during the time period [1, 1.001] is approximately 0.13 cm/sec.