Suppose a certain object moves in a straight line with velocity v(t)= -2+t+3sin(pi t) where v is in meters per second and t is in seconds. Determine the net change in distance of the object from time t=0 to time t=6 seconds and find the object's average velocity on this interval.
Do I just find the antiderivative and solve when t=6? Please help
dx/dt = -2 + t + 3 sin (pi t)
pi t = 2 pi when t = 2 seconds
so we are doing several periods, danger, it luckily asks for the NET distance it goes and not the distance it travels during the 6 seconds (end - begin not total including back and forth)
so yes
x= -2t+t^2/2-(3/pi)cos(pi t)at 6 - at 0
danger again - remember cos 0 = 1
oh, and cos (6 pi) = 1 as well so forget about the trig :)
Thanks so much Damon!
You are welcome.
To determine the net change in distance of the object from time t=0 to time t=6 seconds, you need to find the total distance traveled by the object during this time interval.
To do this, you can integrate the absolute value of the velocity function over the given interval [0, 6] since the distance traveled is always positive.
Let's start by finding the integral of the absolute value of the velocity function:
∫[0,6] |v(t)| dt
To solve this integral, you will need to split it into different intervals where v(t) is positive and negative.
First, let's find the intervals where v(t) is positive:
-2 + t + 3sin(πt) > 0
Simplifying the inequality:
t + 3sin(πt) > 2
Now, solve for t by checking where the inequality is true. You can use graphical or numerical methods to find the intervals where t + 3sin(πt) > 2.
Using a graphical approach, plot the graph of the function y = t + 3sin(πt) and find the regions where it is above the line y = 2.
Once you determine the intervals where the function is positive, you can find the integral of (v(t) = -2 + t + 3sin(πt)) over these intervals.
Similarly, find the intervals where v(t) is negative and then find the integral of (v(t)) over these intervals as well.
The sum of the absolute values of these integrals will give you the total distance traveled by the object from t=0 to t=6 seconds.
To find the object's average velocity on this interval, divide the net change in distance by the total time (6 seconds) to get the average velocity over this interval.