Homework 2 solutions

Problem 49. Setting a world record in a 100 m race, Maggie and Judy cross the finish line in a dead heat, both taking
10.2 s. Accelerating uniformly, Maggie took 2.00 s and Judy 3.00 s to attain maximum speed, which they maintained for the
rest of the race. (a) What was the acceleration of each sprinter? (b) What were their respective maximum speeds? (c) Which
sprinter was ahead at the 6.00 s mark and by how much?
(a,b)
Consider Maggie first. Let P 0 be the Maggie leaving the starting line, P 1 be Maggie finishing her acceleration phase, and
P 2 be Maggie finishing the race.
P 0 P 1 P 2
a ? 0 m/s 2
v 0 m/s ?
x 0 m ? 100 m
t 0 s 2.00 s 10.2 s
Using the 2nd equation from Table 2.2 on page 53 on the first leg:
And again on the second leg:
x 1 = 0.5(v 0 + v 1 )t 1 + x 0 = 0.5v 1 t 1 (10)
x 2 = 0.5(v 2 + v 1 )(t 2 − t 1 ) + x 1 = 0.5(v 1 + v 1 )(t 2 − t 1 ) + 0.5v 1 t 1 = v 1 (t 2 − t 1 + 0.5t 1 ) = v 1 (t 2 − 0.5t 1 ) (11)
x 2
100 m
v 1 = =
= 10.9 m/s
t 2 − 0.5t 1 10.2 s − 0.5 · 2.00 s
(12)
Which is the answer for Maggie in part (b). So
Which answers Maggie in part (a).
Applying the formulas to Judy,
(c)
v 1 =
v 1 = a 1 2t 1 + v 0 (13)
a 1 2 = v 1 /t 1 =
x 2
t 2 − 0.5t 1
=
a 1 2 = v 1 /t 1 =
10.9 m/s
2.00 s
= 5.43 m/s 2 (14)
100 m
= 11.5 m/s (15)
10.2 s − 0.5 · 3.00 s
11.5 m/s
3.00 s
= 3.83 m/s 2 (16)
x M (t) = v M1 (t − t M1 ) + x M1 (17)
x J (t) = v J1 (t − t J1 ) + x J1 (18)
∆x = x M (6 s) − x J (6 s) = 10.9 m/s · 4.00 s + 0.5 · 10.9 m/s · 2.00 s − 11.5 m/s · 3.00 s − 0.5 · 11.5 m/s · 3.00 s = 2.62 m
(19)
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