A swimmer enters a straight river at Point A and attempts to swim directly across to the opposite shore. Instead, the swimmer arrives at Point B diagonally downstream on the opposite shore after 80 seconds of swimming. If the river is 30 m wide and flows downstream at a velocity of 0.5 m/s, what is the minimum distance between Points A and B? (Note: Assume the swimmer does not act to oppose the flow of the river.)

Question

A swimmer enters a straight river at Point A and attempts to swim directly across to the opposite shore.  Instead, the swimmer arrives at Point B diagonally downstream on the opposite shore after 80 seconds of swimming.  If the river is 30 m wide and flows downstream at a velocity of 0.5 m/s, what is the minimum distance between Points A and B?  (Note: Assume the swimmer does not act to oppose the flow of the river.)

Quizplus · Accepted Answer

11ecba2d_1331_860c_a3bf_1b4c8265bf77_MD0008_00 Scalar quantities refer to physical properties that are characterized by only size or magnitude (eg, time, weight). Conversely, vector quantities are characterized by both a magnitude and direction (eg, velocity). Therefore, a vector (V) oriented on an xy-coordinate system is used when an expressing a vector quantity within two-dimensional space: 11ecba2d_1331_860d_a3bf_7d01c43a9da1_MD0008_00 V=x,y Due to this shared convention, the x- and y-components of two vectors V₁ and V₂ may be added together such that the coordinates of the resulting vector (V_R) are expressed as: 11ecba2d_1331_860e_a3bf_55bc2767531b_MD0008_00 11ecba2d_1331_860f_a3bf_2382215e975f_MD0008_00 Furthermore, the magnitude of any vector 11ecba2d_1331_ad20_a3bf_0fd3a7ff65e1_MD0008_00 is equal to the square root of the sum of the x- and y-components squared: 11ecba2d_1331_ad21_a3bf_e1fd4a933e95_MD0008_00 If Point A along the river's shore is set as the origin, the width of the river can be oriented along the y-axis of an xy-coordinate system. Consequently, the vector corresponding to the swimmer's effortful crossing (V_swim) is: 11ecba2d_1331_ad22_a3bf_cf756e99b661_MD0008_00 The flow of the river acts to passively transport the swimmer down the river in the horizontal (x-axis) direction. As such, the swimmer's motion due to the river flow (V_river) may be expressed in terms of the velocity of river flow multiplied by the time it takes the swimmer to cross the river: 11ecba2d_1331_ad23_a3bf_61844b846f5e_MD0008_00 11ecba2d_1331_ad24_a3bf_4390e60f2d00_MD0008_00 Consequently, the vector corresponding to the total motion of the swimmer (V_cross) is the sum of the components of the swimmer's motion along both the x- and y-axes: 11ecba2d_1331_ad25_a3bf_0f1e8dab5e1e_MD0008_00 11ecba2d_1331_d436_a3bf_eb14e10faf9a_MD0008_00 Furthermore, 11ecba2d_1331_d437_a3bf_6b6a2a92eabc_MD0008_00 11ecba2d_1331_d438_a3bf_7570090afe6d_MD0008_00 11ecba2d_1331_d439_a3bf_abb699d1ed94_MD0008_00 Therefore, the distance between Points A and B on opposites sides of the river is at least 50 m (Choice C). Educational objective: Vectors are used to express physical quantities characterized by both a magnitude and a direction. Vectors may be added to one another such that a resultant vector is formed with unique magnitude and direction.

A Swimmer Enters a Straight River at Point a and Attempts