Answer:
(a) See attachment.
(b) The race is a tie.
(c) 20/3 m/s = 6.67 m/s
(d) 400 m
(e)(i) Kim
(e)(ii) Sasha
Step-by-step explanation:
Part a
A speed-time graph is a graphical representation that shows how an object's speed changes over time:
- The horizontal axis represents time (t), measured in seconds (s).
- The vertical axis represents speed (v), measured in metres per second (m/s).
Sasha (blue line)
Sasha accelerates from rest to 8 m/s in 6 seconds. So, the starting point on the speed-time graph is (0, 0). After 6 seconds, her speed is 8 m/s, represented by the point (6, 8). Draw a line from (0, 0) to (6, 8).
Sasha maintains the speed of 8 m/s speed for the next 40 s. So, at 6 + 40 = 46 s, her speed is 8 m/s, represented by point (46, 8). Draw a horizontal line from (6, 8) to (46, 8).
Sasha uniformly decelerates at 4/7 m/s² from a speed of 8 m/s until she stops. Therefore, the time it takes for Sasha to stop is:
[tex]\Delta t=\dfrac{\Delta v}{a}=\dfrac{-8\textsf{ m/s }}{-\frac{4}{7}\textsf{ m/s$^2$ }}=14\;\sf s[/tex]
So, at 46 + 14 = 60 s, Sasha's speed is 0 m/s. Draw a line from point (46, 8) to point (60, 0).
Kim (red line)
Kim accelerates from rest to 8 m/s in 4 seconds. So, the starting point on the speed-time graph is (0, 0). After 4 seconds, her speed is 8 m/s, represented by the point (4, 8). Draw a line from (0, 0) to (4, 8).
Kim maintains the speed of 8 m/s speed until 44 seconds have passed, represented by point (44, 8). Draw a horizontal line from (4, 8) to (44, 8).
Kim uniformly decelerates at 1/2 m/s² from a speed of 8 m/s until she stops. Therefore, the time it takes for Kim to stop is:
[tex]\Delta t=\dfrac{\Delta v}{a}=\dfrac{-8\textsf{ m/s }}{-\frac{1}{2}\textsf{ m/s$^2$ }}=16 \;\sf s[/tex]
So, at 44 + 16 = 60 s, Kim's speed is 0 m/s. Draw a line from point (44, 8) to point (60, 0).
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Part b
Since Sasha and Kim both take 60 seconds to complete the race, it is a tie.
[tex]\dotfill[/tex]
Part c
The average speed can be calculated by dividing the total distance travelled by the total time taken.
The area under the speed-time graph represents the total distance travelled. We can calculate the area under each competitor's graph by breaking the area into simple shapes (triangles and rectangles), calculating the area of each shape, and then summing these areas.
The total distance Sasha travels is:
[tex]d_{\textsf{Sasha}}=\dfrac12\cdot 6\cdot 8+40\cdot 8+\dfrac12\cdot 14\cdot 8 \\\\\\d_{\textsf{Sasha}}=24+320+56\\\\\\d_{\textsf{Sasha}}=400\;\sf m[/tex]
The total distance Kim travels is:
[tex]d_{\textsf{Kim}}=\dfrac12\cdot 4\cdot 8+40\cdot 8+\dfrac12\cdot 16\cdot 8 \\\\\\d_{\textsf{Kim}}=16+320+64\\\\\\d_{\textsf{Kim}}=400\;\sf m[/tex]
Both competitors took 60 seconds to complete the 400 m race, so the average speed of both competitors is:
[tex]\textsf{Average speed}=\dfrac{400}{60}=\dfrac{20}{3}\;\sf m/s[/tex]
[tex]\dotfill[/tex]
Part d
We have already calculated that the distance of the race is 400 m.
[tex]\dotfill[/tex]
Part e (i)
Sasha covers 24 m in the acceleration phase during the first 6 seconds, so t₁ = 6 s. The next 320 m are covered in the constant speed phase, so we need to calculate the time it takes to cover 76 m in this phase, since 100 - 24 = 76. To do this, divide the remaining distance (76 m) by the speed (8 m/s):
[tex]t_2=\dfrac{76}{8}=9.5\;\sf s[/tex]
Therefore, the time it takes Sasha to cover 100 m is:
[tex]\textsf{100m}_{\textsf{Sasha}}=t_1+t_2=6+9.5=15.5\;\sf s[/tex]
Kim covers 16 m in the acceleration phase during the first 4 seconds, so t₁ = 4 s. The next 320 m are covered in the constant speed phase, so we need to calculate the time it takes to cover 84 m in this phase since 100 - 16 = 84. To do this, divide the distance (84 m) by the speed (8 m/s):
[tex]t_2=\dfrac{84}{8}=10.5\;\sf s[/tex]
Therefore, the time it takes Kim to cover 100 m is:
[tex]\textsf{100m}_{\textsf{Sasha}}=t_1+t_2=4+10.5=14.5\;\sf s[/tex]
So, Kim was first in the race after 100 m, as it took her less time than Sasha to cover 100 m.
Part e (ii)
Sasha covers 24 m in the acceleration phase within the first 6 seconds, so t₁ = 6 s.. The time it takes to cover the remaining 276 m at 8 m/s during the constant speed phase is:
[tex]t_2=\dfrac{276}{8}=34.5\;\sf s[/tex]
Therefore, the time it takes Sasha to cover 300 m is:
[tex]\textsf{300m}_{\textsf{Sasha}}=t_1+t_2=6+34.5=40.5\;\sf s[/tex]
Sasha covers 16 m in the acceleration phase within the first 4 seconds, so t₁ = 4 s.. The time it takes to cover the remaining 284 m at 8 m/s during the constant speed phase is:
[tex]t_2=\dfrac{284}{8}=35.5\;\sf s[/tex]
Therefore, the time it takes Kim to cover 300 m is:
[tex]\textsf{300m}_{\textsf{Sasha}}=t_1+t_2=4+35.5=39.5\;\sf s[/tex]
So, Sasha was first in the race after 300 m, as it took her less time than Kim to cover 300 m.
