Question

Race: Blythe and Geoff compete in a 1-km race. Blythe’s strategy is to run the first 600 m of the race at a constant speed of 4 mm/ss, and then accelerate with constant acceleration to her maximum speed of 7.5 mm/ss, which takes her 1 min, and finish the race at that speed. Geoff decides to accelerate with constant acceleration to his maximum speed of 8 m/s at the start of the race and to maintain that speed throughout the rest of the race. It takes Geoff 3 min to reach his maximum speed. Who wins the race?

Answer 1

Answer:

From the value of each person's time to finish the race, it can be concluded that Geoff will win the race.

Explanation:

Given;

total distance to be covered, d = 1000 m

first distance covered by Blythe, $d_b_1$ = 600 m

constant speed of Blythe at the first stage, $v_b_1$ = 4 m/s

final speed of Blythe, $v_b_f$ = 7.5 m/s

time of Blythe's constant acceleration, $t$ = 1 min = 60 s

initial speed of Geoff, $v_g_1$ = 0

final speed of Geoff, $v_g_f$ = 8 m/s

time of Blythe's constant acceleration, $t$ = 3 min = 180 s

The acceleration of Blythe is given as;

$a_b = \frac{dv}{dt} = \frac{7.5 - 4}{60} = 0.0583 \ m/s^2$

distance covered by Blythe during this acceleration is given as;

$d_{a_b} = v_b_1t + \frac{1}{2} a_b t^2\\\\d_{a_b} = 4*60 + \frac{1}{2}(0.0583) (60)^2\\\\d_{a_b} = 344.94 \ m$

The remaining distance covered by 7.5 m/s is given as;

$d_f = d - (600 + 344.94)\\\\d_f = 1000- (944.94)\\\\d_f = 55.06 \ m$

The total time for Blythe to covere the entire distance = time taken to cover 600 m + time taken to cover 344.94 m + time taken to cover 55.06 m

$t_b = \frac{600 \ m}{4 \ m/s}\ +\ 60 s \ + \ \frac{55.06 \ m}{7.5 \ m/s} \\\\t_b = 150 s \ + \ 60s \ + \ 7.34 s\\\\t_b = 217.34 s$

The acceleration of Geoff  is given as;

$a_g = \frac{dv}{dt} = \frac{8 -0}{180} = 0.044 \ m/s^2$

distance covered by Geoff during this acceleration is given as;

$d_{a_g} = v_g_1 + \frac{1}{2}a_gt^2\\\\ d_{a_g} =0 + \frac{1}{2}(0.044)(180)^2\\\\ d_{a_g} = 712.8 \ m$

The remaining distance covered by 8 m/s is given a;

$d_f = 1000 \ m - 712.8 \ m\\\\d_f = 287.2 \ m$

The total time for Geoff to covere the entire distance = time taken to cover 712.8 m + time taken to cover 287.2 m

$t_g = 180s + \frac{287.2 \ m}{8 \ m/s} \\\\t_g = 180s + 35.9 s\\\\t_g = 215.9 s$

Therefore, from the value of each person's time to finish the race, it can be concluded that Geoff will win the race