Relative Velocity Question?

Question

A Swimmer is training in a river. The current flows at 1.45 m /s and the swimmer swims upstream a distance of 146 metres before swimming back to the starting point. If the total time for the swim is 189.0 seconds, what is the swimmers speed relative to the water? .... help.... quick.... :)

Answer 1

The solution can be computed from the expression of time of travel. Let us assume that the swimmer has not taken rest during the swimming, i.e. he swimmed continuously. Then, basically the time 189 seconds represents the voyage time.

i.e. 189 = Upstream time (t_u) + downstream time (t_d) .. (1)

Now, the swimmer travels both upstream and downstream to make the round trip. Hence the velocity of the swimmer will not be same in both the cases. He will be moving with different velocity. The actual velocity of the swimmer is given by the vectorial addition of the velocities of river & swimmer.

In upstream river opposes the swimmer movement. The upstream velocity is given by,

V_u = Swimmer Velocity (V_s) - river velocity ( V_r) .... (2a)
= V_s - 1.45 m/s .... (2b)

The time taken by the swimmer to cover 146 m would be given by

t_u = distance / Velocity ... (3a)
= 146 / V_u ... (3b)

In downstream movement river is in the same direction of the swimmer. The upstream velocity is given by,

V_d = V_s - 1.45 m/s ... (4)


The time taken for downstream movement is given by,

t_d = 146/ V_d ... (5)

Now, finally substuiting (3) and (5) in (1)

189 = 146 / V_u + 146 / V_d ... (6a)
= 146 / (V_s - 1.45) + 146/(V_s + 1.45) ... (6b)

The above equation would result in a quadratic equation and solving for V-s will give two solution: 2.4514, -0.8704.

Taking the positive roots, we get

V_u = 0.9654 m/s
V_d = 3.8654 m/s

Answer 2

While going upstream, his velocity is less because of the current. While returning downstream, he is aided by the current.

Upstream : 146 / v - 1.45 is the time in seconds if the swimmer's velocity is v.

Downstream: 146 / v + 1.45 is the time in seconds

Total is 189 seconds.

If we want, we can solve for v by

292/189 = 146 / v -1.45 + 146 / v + 1.45 or

2/189 = 1/v - 1.45 + 1 / v + 1.45

v is the velocity w.r.t. the land.

However, the speed of the swimmer with respect to water is given by 292/189 = 1.545 m/s

Answer 3

1.545 ms^-1