Question

A river has a steady speed of 0.650 m/s. A student swims upstream a distance of 1.00 km and swims back to the starting point.

(a) If the student can swim at a speed of 1.15 m/s in still water, how long does the trip take? s

(b) How much time is required in still water for the same length swim? s

(c) Intuitively, why does the swim take longer when there is a current?

Answer 1

What happens if the water

is

still? The student swims a distance of 1.00 km “upstream”

at a speed of 1.15 m/s

using the simple distance/time formula

d = vt

the time for the trip is

t = d/v = 1.0×10^3m/1.15 m/s = 1000/1.15= 870

and the same is true for the trip back “downstream”. So the tot

at time for the trip is

870 s + 870 s = 1740 s = 1.74*10^3 s = 29 min

the velocity of the water with respect to the groundis

vWG = -0.650 m/s

velocity with respect to the water is

vSW= +1.15m/s.

the water and the water’s velocity with respect to the ground:

vSG=vSW+vWG = 1.15m/s-0.650m/s =0.5m/s

displacement of ∆x=1.00km

delta t=delta x/delta SG =1.00*10^3 m/0.5m/s = 2000 s

Then the student swims downstream and his velocity with respect to the water is

vSW=-1.15m/s.

velocity with respect to the ground of

vSG = vSW+vWG = -1.15m/s-0.650 m/s = -1.8m/s

time to cover a displacement of ∆x=−1.00 km is

∆t=∆x/vSG=(-1.00*10^3m)/(-1.8m/s) =555.55 s

the total time to swim upstream and then downstream is

tTotal=tup+tdown = 2000s+555.55s= 42.59 min

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(b) If the water were still the distance is

1000m/1.15m/s =869.56 s

total time is

869.56*2 = 1739.12 s
the time required in still water for the same length swim is

2555.55-1739.12 = 816.43 s = 13.6 min
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(c) when there is a current the student swims equal distances moving slowlyand moving rapidly realtive to the shore. but his average speed is the time he spends with each different speed, consider here he spends more than twice as much time with the lower spped, so his average speed is lower than it would be without a current. this depressed average speed makes the round-trip time longer when the current is flowing.