The distance between two points is the number of units between them.
The cart move the greatest distance in segment AB
Calculate distance AB
To do this, we calculate the perimeter of the quarter of the ellipse
This is calculated using:
[tex]\mathbf{P = \frac{\pi}{4} (a + b) ( 3 \times \frac{(a b)^2}{(a+b)^2(\sqrt{-3\frac{(ab)^2}{a + b)^2 }+ 4}\ + 10} + 1}) }[/tex]
Where:
[tex]\mathbf{a = 4ms^{-1}}[/tex]
[tex]\mathbf{b = 0.6s}[/tex]
Using a calculator,
[tex]\mathbf{P = 4.13m}[/tex]
So:
[tex]\mathbf{AB = 4.13m}[/tex]
Calculate distance BC
From the graph (see attachment), we have:
[tex]\mathbf{time = 0.2s}[/tex]
[tex]\mathbf{velocity = 8ms^{-1}}[/tex]
So, the distance traveled is:
[tex]\mathbf{distance = time \times velocity}[/tex]
[tex]\mathbf{distance = 0.2s \times 8ms^{-1}}[/tex]
[tex]\mathbf{distance = 1.6m}[/tex]
So:
[tex]\mathbf{BC = 1.6m}[/tex]
Calculate distance CD
From the graph, we have:
[tex]\mathbf{u = 8}[/tex] --- initial velocity at point C
[tex]\mathbf{v = 0}[/tex] ---- final velocity at point D
[tex]\mathbf{t = 0.2}[/tex] --- time of travel
So, the acceleration is:
[tex]\mathbf{a = \frac{v - u}{t}}[/tex]
[tex]\mathbf{a = \frac{0 - 8}{0.2} = -40ms^{-2}}[/tex]
The distance travelled is:
[tex]\mathbf{s = ut + \frac{1}{2}at^2}[/tex]
So, we have:
[tex]\mathbf{s = 8 \times 0.2 - \frac{1}{2} \times 40 \times 0.2^2}[/tex]
[tex]\mathbf{s = 1.6 - 0.8}[/tex]
[tex]\mathbf{s = 0.8}[/tex]
So:
[tex]\mathbf{CD = 0.8m}[/tex]
Calculate distance DE
From the graph, we have:
[tex]\mathbf{u = 0}[/tex] --- initial velocity at point D
[tex]\mathbf{v = -4}[/tex] ---- final velocity at point E
[tex]\mathbf{t = 0.6}[/tex] --- time of travel
So, the acceleration is:
[tex]\mathbf{a = \frac{v - u}{t}}[/tex]
[tex]\mathbf{a = \frac{-4 - 0}{0.6} = -6.67ms^{-2}}[/tex]
The distance travelled is:
[tex]\mathbf{s = ut + \frac{1}{2}at^2}[/tex]
So, we have:
[tex]\mathbf{s = 0 \times 0.6 - \frac{1}{2} \times 6.67 \times 0.6^2}[/tex]
[tex]\mathbf{s = 0 - 1.2}[/tex]
[tex]\mathbf{s = - 1.2}[/tex]
So:
[tex]\mathbf{DE = 1.2m}[/tex]
By comparing the distances, segment AB is the longest i.e. 4.13 m
Hence, the cart move the greatest distance in segment AB
Read more about distance and rates at:
https://brainly.com/question/21149231
