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 simultane- ously, both setting a world record of 10.4 s

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