(a) Explanation Given information: The half-length ( Δ x ) of the race is 200 m . Runner A reaches her maximum velocity ( v A ) in time ( t 1 ) A 4 sec. Runner A reaches the half waypoint with split time ( t 2 ) A of 25 sec. Runner B reaches her maximum velocity ( v B ) in time ( t 1 ) B 5 sec. Runner B reaches the half waypoint with split time ( t 2 ) B of 25.2 sec. Both runner A ( a A ) and runner B ( a B ) has constant deceleration of 0.1 m/s 2 . Calculation: Tabulate the velocity ( v ) corresponding to time ( t ) for runner A as in Table (1). t (s) Runner A v ( m/s ) 0 0 4 ( v A ) 25 ( v A ) Plot the v-t curve from runner A with area as in Figure (1). Tabulate the velocity ( v ) corresponding to time ( t ) for runner B as in Table 2. t (s) Runner B v ( m/s ) 0 0 5 ( v B ) 25.5 ( v B ) Plot the v-t curve from runner B with areas as in Figure 2. Calculate the area ( A 1 ) using the relation: A 1 = 1 2 [ ( t 1 ) A × v A ] Substitute 4 sec for ( t 1 ) A . A 1 = 1 2 [ 4 × v A ] = 2 v A (1) Calculate the area ( A 2 ) using the relation: A 2 = ( ( t 2 ) A − ( t 1 ) A ) v A Substitute 4 sec for ( t 1 ) A and 25 sec for ( t 2 ) A . A 2 = ( 25 − 4 ) × v A = 21 v A (2) Calculate the maximum velocity ( v A ) of runner A : Add Equations (1) and (2). A 1 + A 2 = 2 v A + 21 v A = 23 v A Equate ( Δ x ) to ( A 1 + A 2 ) Δ x = ( A 1 + A 2 ) Substitute 200 m for Δ x and 23 v A for ( A 1 + A 2 ) . 200 = 23 v A v A = 200 23 v A = 8.6957 m/s Calculate the area ( A 3 ) using the relation: A 3 = 1 2 [ ( t 1 ) B × v B ] Substitute 5 sec for ( t 1 ) B . A 3 = 1 2 [ 5 × v B ] = 2.5 v B (3) Calculate the area ( A 4 ) using the relation: A 4 = ( ( t 2 ) B − ( t 1 ) B ) v B Substitute 5 sec for ( t 1 ) A and 25.2 sec for ( t 2 ) A . A 2 = ( 25.2 − 5 ) × v B = 20.2 v B (4) Calculate the maximum velocity ( v B ) of runner B : Add Equations (3) and (4). A 3 + A 4 = 2 (b) To determine The position of the winner ( Δ x ) relative to the loser when the winner reaches the finish line.

12th Edition

ISBN: 9781260663778

Author: BEER

Publisher: MCG

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Chapter 11.3, Problem 11.71P

(a)

To determine

The race times for runner A (tA) and runner B (tB).

(b)

To determine

The position of the winner (Δx) relative to the loser when the winner reaches the finish line.

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Chapter 11.3, Problem 11.70P

Chapter 11.3, Problem 11.72P

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The race times for runner A ( t A ) and runner B ( t B ) .

Solution Summary: The author calculates the velocity ( v ) corresponding to time for runner A as in Table (1).