Relative Velocity of Swimmer: How to Calculate Total Time for Different Paths?

In summary, the conversation involved discussing calculations for the time and velocity of swimmer movements. Diagrams were used to aid in the calculations and the final result for the time difference, ΔT, was given in terms of the time for part (a), Ta. There were some minor errors in the equations but the overall results appeared to be correct.
  • #1
Taylor_1989
402
14
Homework Statement
The figure below depicts swimmers A and B who a race. A swim directly from the start point P to point Y (a distance L) and back again. B swims in a direction orthogonal to A from the start point P to point X (also a distance L) and back again. In the absence of any currents, the two swimmers have the same top speed of c. Both swimmer maintain a perfectly straight line through their respective courses and exert a maximum effort throughout.

a) Assume that there is no current. How long dose the journey for each swimmer take?

b) Let us now assume that a current flows with speed v in the negative y direction as shown on the figure. By consideration of the effective component of the velocity vectors along the two directions of travel or otherwise, find an expression for the difference in time of arrival $\Delta t$, of the two swimmers in terms of the time in (a) and the parameter $\beta =v/c$
Relevant Equations
##v=d/t##
##c=\sqrt(a^2+b^2)##
So I was just wondering if someone could check my method for (b) as sometimes I can have a tendency of getting the relative components wrong ect.

Diagram 1

Forum.png


(a)
Time for PY: ##T=L/c##
Time for YP: ##T=L/c##
Total Time:##2L/c##

(b)
Velocity for PY: ##c-v##
Time: ##T=L/(c-v)##

Velocity for YP: ##c+v##
Time : ##T=L/(c+v)##

Total time : ##T_{t1}=2cL/(c^2-v^2)##

Now for PX and XP I used the following diagram 2

diagram-20190409.png


From these I calculated the velocities of PX and XP of the swimmer to be

PX : ##v_s=\sqrt{c^2-v^2}##
XP : ##v_s=\sqrt{c^2-v^2}##

So calculating the total time

##T_{t2}=\frac{2L}{\sqrt{c^2-v^2}}=\frac{2L\cdot \sqrt{c^2-v^2}}{c^2-v^2}##

Next I simplified ##T_{t1}## and ##T_{t2}##

##T_{t1}=\frac{2L}{c\sqrt{1-\beta^2}}##

##T_{t2}=\frac{2L \sqrt{1-\beta^2}}{c(1-\beta^2)}##

so ##\Delta T## is given by

##\Delta T = \frac{2L(\sqrt{1-\beta^2})}{c(1-\beta^2)}##
 
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  • #2
Diagram 2 appears to be rotated by 90 degrees relative to diagram 1. But that doesn't affect your results. Your results for ##T_{t1}## and ##T_{t2}## look correct to me. However, there are some "typos" at the end where you wrote:
Taylor_1989 said:
Next I simplified ##T_{t1}## and ##T_{t2}##

##T_{t1}=\frac{2L}{c\sqrt{1-\beta^2}}##

##T_{t2}=\frac{2L \sqrt{1-\beta^2}}{c(1-\beta^2)}##

so ##\Delta T## is given by

##\Delta T = \frac{2L(\sqrt{1-\beta^2})}{c(1-\beta^2)}##

Did you write ##T_{t1}## correctly here? Note that your expressions for ##T_{t1}## and ##T_{t2}## are equivalent.

What should ##\Delta T## reduce to if ##v = 0##? Does your result do this?

You are asked to express ##\Delta T## in terms of the time ##T_a## for part (a).
 

1. What is relative velocity of a swimmer?

The relative velocity of a swimmer is the speed and direction at which the swimmer is moving in relation to another object or point of reference. It takes into account the swimmer's own speed and direction, as well as the speed and direction of the object or point of reference.

2. How is relative velocity of a swimmer calculated?

The relative velocity of a swimmer can be calculated using vector addition. This involves adding the swimmer's velocity vector to the velocity vector of the object or point of reference. The resulting vector represents the relative velocity of the swimmer.

3. Why is relative velocity of a swimmer important?

Relative velocity of a swimmer is important because it helps to understand how the swimmer is moving in relation to other objects or points of reference in the water. This can be useful for making strategic decisions in a race or for analyzing the swimmer's performance.

4. How does relative velocity of a swimmer affect their performance?

The relative velocity of a swimmer can affect their performance in a number of ways. If the swimmer is moving in the same direction as the current or another object, their relative velocity will be higher and they may be able to swim faster. However, if they are moving against the current or another object, their relative velocity will be lower and they may have to work harder to maintain their speed.

5. Can relative velocity of a swimmer change?

Yes, the relative velocity of a swimmer can change depending on their own speed and direction, as well as the speed and direction of the object or point of reference. This can change as the swimmer moves through the water or as the current or other objects in the water change.

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