9*B A bug crawls in a straight line according to the velocity
graph at right where velocity is in meters per second.
VO
a.
Explain what is happening when 2 st 54.
b.
Calculate the total distance the bug traveled.
How far does the bug end up from its starting point?
d.
When is the bug's velocity:
i. 0
> 0
<0
e.
When is the bug's acceleration:
i. 0
> 0
<0
f.
At what time does the bug pass over its starting point?

a. The bug is moving at a constant speed of -2 m/s

b. The total  distance traveled by the bug is 17 meters

c. The distance the bug end up from its starting point is 3 meters

d. i. The bugs velocity is 0 when t = 0, 5, or 10 seconds,

ii. The bugs velocity is > 0 at 5 ≤ t ≤ 10

iii. The bugs velocity is < 0 at 0 ≤ t ≤ 5

e. i. The bug's acceleration is 0 at 2 ≤ t ≤ 4

ii. The bug's acceleration is > 0 at 4 ≤ t ≤ 6 and 8 ≤ t ≤ 9

iii. The bug's acceleration is > 0 at 0 ≤ t ≤ 2 and 9 ≤ t ≤ 10

f. The time the bug passes the starting point is approximately 8.83 seconds

Explanation:

a. When 2 ≤ t ≤ 4, the bug is moving at a constant speed of -2 m/s

b. The total distance the bug traveled is given by the area under the velocity-time graph which consist of two trapezoid and a triangle and is therefore is given as follows;

The area of the first (negative) trapezoid = -2 × (2 + 5)/2 = -7

The area of the second trapezoid = 2 × (3.5 + 5)/2 = 8.5

The area of the triangle = 1/2 × 1.5 × 2 = 1.5

The sum of the magnitude of the areas = The total  distance traveled by the bug = + 8.5 + 1.5 = 17

The total  distance traveled by the bug = 17 meters

c. The distance the bug end up from its starting point is given by the sum of the displacement (total displacement) of the bug

The total displacement of the bug = -7 + 8.5 + 1.5 = 3

The distance the bug end up from its starting point = The total displacement of the bug = 3 meters

d. i. The bugs velocity is 0 when the graph crosses the time, 't', axis, which is at points for time, t = 0 seconds, t = 5 seconds, and t = 10 seconds

ii. The bugs velocity is > 0 between the times, 5 ≤ t ≤ 10

iii. The bugs velocity is < 0 between the times, 0 ≤ t ≤ 5

e. i. The bug's acceleration is 0 between the times, 2 ≤ t ≤ 4

ii. The bug's acceleration is > 0 between the times, 4 ≤ t ≤ 6 and 8 ≤ t ≤ 9

iii. The bug's acceleration is > 0 between the times, 0 ≤ t ≤ 2 and 9 ≤ t ≤ 10

f. The bug passes the starting point at the time, 8 + t, where 't', is given  by the area of a trapezoid as follows;

2 = t × (2 +  2 + t)/2

∴ 4 = 4·t + t²

t² + 4·t - 4 = 0

(t + (2 + 2·√2))·(t - (2·√2 - 2))

∴ t = 2·√2 - 2

The time the bug passes the starting point = 8 + 2·√2 - 2 = 6 + 2·√2 ≈ 8.83

The time the bug passes the starting point ≈ 8.83 seconds

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