Confusion on relative velocity sign

[niko_niko]

Homework Statement

Problem: Two swimmers, Chris and Sarah, start together at the same point on the bank of a wide stream that flows with a speed v. Both move at the same speed c (where c > v) relative to the water. Chris swims downstream a distance L and then upstream the same distance.

Relevant Equations

v_PA = v_PB + u_BA

The downstream part is no problem: c_E = c + v.
My only question is why is the upstream velocity of Chris relative to the Earth c-v, as said by the textbook, and not v-c? Assuming v is to the right and positive, doesn't c become negative since it is in the opposite direction?

 

[PeroK]

Homework Statement: Problem: Two swimmers, Chris and Sarah, start together at the same point on the bank of a wide stream that flows with a speed v. Both move at the same speed c (where c > v) relative to the water. Chris swims downstream a distance L and then upstream the same distance.
Relevant Equations: v_PA = v_PB + u_BA

The downstream part is no problem: c_E = c + v.
My only question is why is the upstream velocity of Chris relative to the Earth c-v, as said by the textbook, and not v-c? Assuming v is to the right and positive, doesn't c become negative since it is in the opposite direction?

In terms of velocity, if downstream is positive, then upstream is negative. If you change the sign of $c$, then you are changing the positive direction.

 

[PeroK]

The upstream speed is, of course, $v -c$, as speed is non-negative.

 

[jbriggs444]

The upstream speed is, of course, $v -c$, as speed is non-negative.

For some reason, the current speed is $v$ while the swimmer speed is $c$. With $c > v$, I make the upstream speed $c-v$.

Authors would be well advised to pick mnemonic variable names.

If the textbook uses the phrase "upstream velocity", this could be interpreted to indicate the use of an upstream-positive coordinate system. So the positive value $c-v$ would be appropriate.