the answer was negative since the part of the curve was under the x-axis (or t-axis) was larger than the part above it. for distance travelled however, we cannot have a negative value. so we must find these two areas in terms of magnitude and add them together. to find the areas as positive maginitudes i needed to split the integral in 2 and turn the limits of the first backward. that gives me positive areas for both pieces
The position function is: .s(t) .= .∫(3t - 5) dt .= .3t²/2 - 5t + Cthe velocity function (in m/sec) is given for a particle moving along a line.
. . v(t) .= .3t - 5, .0 < t < 3
(a) Find displacement
(b) Distance traveled
(a) When t = 0: .s(0) .= .3·0²/2 - 5·0 + C .= .C
. . .When t = 3: .s(3) .= .3·3²/2 - 5·3 + C .= .C - 3/2
The displacement is: .(C - 3/2) - C .= .-3/2
(The particle is 1½ units to the left of where it started.)
(b) The velocity is zero when t = 5/3
. . .(That's when the particle changes direction.)
We know that: .s(0) = C
When t = 5/3: .s(5/3) .= .C - 25/6
. . The displacement is: .(C - 25/6) - C .= .-25/6
It has moved 25/6 units to the left.
When t = 3: .s(3) .= . C - 3/2
. . The displacement is: .(C - 3/2) - (C - 25/6) .= .8/3
It has moved 8/3 units to the right.
Therefore, the total distance is: .25/6 + 8/3 .= .41/6 units.