Complete question is;
In a women’s 100-m race, accelerating uniformly, Laura takes 2.00 s and Healan 3.00 s to attain their maximum speeds, which they each maintain for the rest of the race. They cross the finish line simultaneously, both setting a world record of 10.4 s.
(a) What is the acceleration of each sprinter?
(b) What are their respective maximum speeds?
(c) Which sprinter is ahead at the 6.00-s mark, and by how much?
(d) What is the maximum distance by which Healan is behind Laura, and at what time does that occur?
Answer:
A) a_laura = 5.319 m/s² and a_healan = 3.745m/s²
B) V_max for laura = 10.638 m/s and V_max for Healan = 11.235 m/s
C) Laura is ahead of healan by 2.632 m
D) Healan is 4.471 m behind Laura
Explanation:
A) The distance traveled by each sprinter with constant acceleration would be gotten from the equation of motion which is;
s = ut + ½at²
Now, since they began at rest, thus, initial velocity(u) = 0 m/s
So, s = 0 + ½at²
s = ½at²
Now, maximum speed achieved by each of them is given by the formula;
v_max = u + at
Since u = 0, then;
v_max = at
Now, we are told that they set a world record of 10.4 s, thus;
Time taken by each to run at their maximum speed = (10.4 - t)
Distance travelled after attaining maximum speed = at + a(10.4 - t)
Thus;
Total distance covered by each sprinter = ½at² + (at)(10.4 - t)
Where;
t = time taken to accelerate to maximum speed
Since total distance covered is 100 m and she took 2 s to accelerate to max speed, then Acceleration for Laura would be calculated from;
100 = ½a(2)² + (2a(10.4 - 2))
This gives;
100 = 2a + 16.8a
18.8a = 100
a = 100/18.8
a_laura = 5.319 m/s²
Similarly, for Healan:
100 = ½a(3²) + (3a(10.4 - 3)
100 = 4.5 a + 22.2a
26.7a = 100
a = 100/26.7
a_healan = 3.745m/s²
B) From earlier we saw that max speed is;
v_max = at
Thus, for laura;
Vmax for laura= 5.319 × 2
V_max for laura = 10.638 m/s
Also, for healan;
Vmax = 3.745 × 3
V_max for Healan = 11.235 m/s
C) to find which sprinter is ahead at 6.00 s,
For Laura:
D = (½(5.319 × 2²)) + (2² × 10.638)
D_laura = 53.19 m
For Helen: D = (½(3.745 × 3²)) + (3 × 11.235) = 50.558 m
Difference = 53.19 - 50.558 = 2.632
Thus, Laura is ahead by 2.632 m
C) We are told that laura achieved max speed in 2 s while healan took 3 s. This means that when Laura had reached her maximum speed Healan was still accelerating. Thus;
From v_max = at
10.638 = (3.745)t
t = 10.638/3.745
t = 2.84 s
Distance travelled during this time is;
For Laura:
D = (½(5.319 × 2²) + (0.84)(10.638) = 19.574 m
Foe Healan:
D = (½(3.745 × 2.84²)) = 15.103 m
Difference = 19.574 - 15.103 = 4.471 m
Thus, Healan is 4.471 m behind laura
