You can do the math for this. I dont have time right now. (1/3(3)^3 - (3)^2 - 3(3) + 4) - ((1/3)(0)^2 - (0)^2 - 3(0) + 4) = ? Do the same thing for 6 and subtract it from 3. Add the two values together and that is your total distance.(1/3(6)^3 - (6)^2 - 3(6) + 4) - ((1/3)(3)^2 - (3)^2 - 3(3) + 4) = ?......
At the instant when it changes direction its velocity will = 0
v(t)= t^2 - 2t - 3 = 0
(t -3)(t +1) = 0
velocity changes at t = 3 and t = -1
2) The total distance traveled from 0 < t < 6 is: ______.
particle moves from t = 0 to t = 3 then returns until t = -1
so in 6 sec it moves out for three seconds and then back for 3 seconds
total distance traveled is: 2s(t) = 2*[(1/3)t^3 - t^2 - 3t + 4]
total distance traveled = 2[9 - 9 - 9 +4] = -10
Note that the - sign indicates direction
total displacement = 0
3) The total displacement from 0 < t < 6 is ______.
4) The average velocity from 0 < t < 6 is ______ .
5) The average speed from 0 < t < 6 is ______.
1) Set v(t) equal to 0. Solve for t.
It is a quadratic, so I'll just use a quadratic calculator online.
t1 = -1
t2 = 3
We can throw out t1. Time cannot be negative.
So the particle changes direction at time t = 3.
2) The total distance traveled is how far is travels from 0 to 3 and then from 3 to 6. If you do from 0 to 6, you will only find the displacement since it changes direction.
You can do the math for this. I don't have time right now.
(1/3(3)^3 - (3)^2 - 3(3) + 4) - ((1/3)(0)^2 - (0)^2 - 3(0) + 4) = ?
Do the same thing for 6 and subtract it from 3. Add the two values together and that is your total distance.
(1/3(6)^3 - (6)^2 - 3(6) + 4) - ((1/3)(3)^2 - (3)^2 - 3(3) + 4) = ?
3) Total displacement. Just do the same thing as above but subtract 6 and 0 in the s(t) function.
(1/3(6)^3 - (6)^2 - 3(6) + 4) - ((1/3)(0)^2 - (0)^2 - 3(0) + 4) = ?
4) Average velocity is just the change in distance over the change in time. Take the answer from #3 and divide by 6 seconds.
5) Speed is the same as velocity, but it has no direction. It is sort of in a way the absolute value of Velocity. If you get a negative in #4, just take the absolute value.
position, velocity, and acceleration are derivatives of each other.
ie. position is in distance, velocity (distance/time), and acceleration (distance^2/time)
