Explanation Given: Given that, B’s speed is remained constant. Calculation: Construct the right differential equations for each of A ’s and B ’s velocities as a function of time, v a ( t ) , and v b ( t ) , respectively. Since the velocity of B is constant, so: v b ( t ) = v b 0 First note that the difference between the two drivers was steady at first. Let this time be t t = 0 which means that the two drivers had the same initial velocity. Hence, v a ( 0 ) = v b 0 Where time t = 0 is the time at which driver A started to run out of gas. Since the speed is proportional to square the velocity as shown below, d v a d t = k v a 2 , v a ( 0 ) = v b 0 Where k < 0 Since the speed is decreasing. Separate the variable as follows, d v a d t = k v a 2 d v a v a 2 = k d t Integrate the above equation. ∫ d v a v a 2 = ∫ k d t − 1 v a = k t + C …… ( 1 ) Apply the initial condition. Substitute 0 for t , and v b 0 for v a in equation ( 1 ) . − 1 v b 0 = k ( 0 ) + C C = − 1 v b 0 Substitute − 1 v b 0 for C in equation ( 1 ) . − 1 v a = k t − 1 v b 0 v a = − 1 k t − 1 v b 0 v a = v b 0 1 − v b 0 k t …… ( 2 ) Calculate k , our negative constant. Since the other given condition in terms of distance “One mile later, driver A ’s speed was exactly halved.”Integrate velocity function to get a distance ( x(t) ) function and apply the condition on it, then get k from that equation. Since the relation between v a and x is given as follows, d x d t = v a , x ( 0 ) = 0 , and v a ( t 1 miles ) = 1 2 v b 0 Integrate the equation ( 2 ) as follows, ∫ v a d t = ∫ v b 0 1 − v b 0 k t d t x ( t ) = − 1 k ln ( v b 0 k t − 1 ) + C ' …… ( 3 ) Apply the initial condition x ( 0 ) = 0 in equation ( 3 )

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Chapter 2.2, Problem 39E

To determine

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The winner of the race.

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Ch. 2.2 - Prob. 1E Ch. 2.2 - In Problems 1-6, determine whether the given...Ch. 2.2 - Prob. 3E Ch. 2.2 - Prob. 4E Ch. 2.2 - Prob. 5E Ch. 2.2 - Prob. 6E Ch. 2.2 - Prob. 7E Ch. 2.2 - Prob. 8E Ch. 2.2 - Prob. 9E Ch. 2.2 - Prob. 10E

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The winner of the race.

To find: The winner of the race.

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Chapter 2 Solutions

To find:

The winner of the race.