Step-by-step explanation:
m/s² = m/s / s
this is the slope of the line in the graph.
t = 2, that means it is in the first segment of 0 - 6 seconds
the slope in this segment is
(25-15)m/s / (6-0)s
10m/s / 6s = 10/6 m/s² = 5/4 m/s² = 1.25 m/s² ≈ 1.3 m/s²
the same thing for t = 12 in the third segment.
the acceleration or slope is negative here, but that's called here a "forbidden word", and is therefore also a positive number.
in that this segment the slope is
(0-25)m/s / (16-11)s
-25m/s / 5s = -5 m/s²
and so, the answer is 5 m/s².
for the total distance we need to calculate the area under the lines.
the line functions in the 3 segments are :
s(t) = 5/4 t + 15
s(t) = 25
s(t) = -5t + 25
the third one is simplified by moving it from the 11 to 16 seconds interval to a 0 to 5 seconds interval. this creates the same area under that line.
the area under the first line is the integral of the line function between 0 and 6.
that is
[5/8 t² + 15t] between 0 and 6.
and that is
5/8 × 36 + 90 - 5/8 × 0 - 15×0 = 45/4 + 90
= 45/4 + 360/4 = 405/4 = 101.25m
the area under the second segment is simply the area of the rectangle 5×25 = 125m
and the area under the third segment is the integral of the line function between 0 and 5 (remember, we shifted from 11 to 16 to the interval 0 to 5).
that is
[-2.5t² + 25t] in the interval 0 to 5.
and that is
-2.5×25 + 25×5 - -2.5×0 - 25×0 = -62.5 + 125 = 62.5m
as a control check this area is also the area of a right-angled triangle with legs of 5 and 25
the area is 25×5/2 = 125/2 = 62.5
so, it is correct.
the total area under the lines is therefore
101.25 + 125 + 62.5 = 288.75 m ≈ 288.8 m
the average speed is now simply
288.75 m / 16 s = 288.75/16 m/s = 18.046875 m/s ≈
≈ 18 m/s
but if we used the already rounded result of the total distance of 288.8 m, then we would get
288.8 / 16 = 18.05 m/s ≈ 18.1 m/s
