An object moves along the x-axis with a velocity of v(t) = sin(pi/3)t for time t greater than or equal to 0. What is the total distance traveled by the object over the time interval 0 less than or equal to t less than or equal to 4?
well, from t = 0 to t = 3
sin(pi t/3 ) is positive
and
from t = 3 to t = 4, sin (pi t/3) is negative but the distance is still a positive quantity even if movingbackwards
so
integrate from t = 0 to t = 3
sin(pi t/3)dt
then integrate from t = 3 to t = 4
sin (pi t/3) dt
add the absolute values of both
To find the total distance traveled by the object over the given time interval, we need to calculate the displacement of the object and then take the absolute value of the displacement.
The displacement of an object is given by the integral of its velocity function.
Given that the velocity function v(t) = sin(pi/3)t for t greater than or equal to 0, we can calculate the displacement as follows:
Displacement = ∫v(t) dt from 0 to 4
= ∫sin(pi/3)t dt from 0 to 4
= [-cos(pi/3)t^2/2] from 0 to 4
= -cos(pi/3)(4^2/2) - (-cos(pi/3)(0^2/2))
= -cos(pi/3)(16/2) - (-cos(pi/3)(0))
= -cos(pi/3)(8) - (-cos(pi/3)(0))
Since cos(pi/3) = 1/2, we can simplify this further:
= -(1/2)(8)
= -4
Now, to find the total distance traveled, we take the absolute value of the displacement:
Total Distance = |Displacement|
= |-4|
= 4 units
Therefore, the total distance traveled by the object over the time interval 0 ≤ t ≤ 4 is 4 units.
To find the total distance traveled by the object over the given time interval, we need to integrate the absolute value of the velocity function from 0 to 4.
First, let's find the integral of the absolute value of the velocity function. We have v(t) = sin(pi/3)t.
Step 1: Determine the absolute value of v(t) by taking the absolute value of sin(pi/3)t.
|v(t)| = |sin(pi/3)t|
Step 2: Integrate the absolute value of v(t) from 0 to 4.
Integral(|v(t)|, t=0 to 4) = Integral(|sin(pi/3)t|, t=0 to 4)
To solve this integral, we need to break it into two parts since the function inside the absolute value changes sign.
Part 1: When sin(pi/3)t >= 0 (0 ≤ t ≤ 4)
In this case, |sin(pi/3)t| = sin(pi/3)t
Integral of sin(pi/3)t from 0 to 4:
Integral(sin(pi/3)t, t=0 to 4) = [-cos(pi/3)t/(pi/3)] evaluated from 0 to 4
= (-cos(4pi/3)/(pi/3)) - (-cos(0)/(pi/3))
= [-cos(4pi/3) + 1]/(pi/3)
Part 2: When sin(pi/3)t < 0 (0 ≤ t ≤ 4)
In this case, |sin(pi/3)t| = -sin(pi/3)t
Integral of -sin(pi/3)t from 0 to 4:
Integral(-sin(pi/3)t, t=0 to 4) = [cos(pi/3)t/(pi/3)] evaluated from 0 to 4
= (cos(4pi/3)/(pi/3)) - (cos(0)/(pi/3))
= [cos(4pi/3) - 1]/(pi/3)
Now, we need to sum up the results from Part 1 and Part 2 to find the total distance traveled.
Total distance traveled = [(cos(4pi/3) - 1) - (cos(4pi/3) + 1)]/(pi/3)
= [-2]/(pi/3)
= -6/pi
Therefore, the total distance traveled by the object over the time interval 0 ≤ t ≤ 4 is -6/pi units.