Wording of Relative Velocity problems [closed]

Question

I've been doing some relative velocity problems. The three main types I do are to do with cars, boats/rivers and planes/crosswind.

I'm fine with cars as the wording is pretty straightforward - it is just car a, car b what is relative to what and I can construct that into the three relative velocity equations.

However, I have an issue with boats and rivers: For example, here's a question I got -

"A boat needs to travel upstream to pick up passengers. The boat travels at $10ms^{-1}$, while the downstream current is $2ms^{-1}$. If the boat needs to travel upstream $3000m$, the needs to wait 2 minutes before travelling downstream for $6000m$, how long does the total time take?"

For this question, I assumed that, in still water the boat is $10ms^{-1}$. So the upstream speed is $8ms^{-1}$ and downstream is 10, so I got 16 minutes 35 s overall.

Is this the right assumption if the question doesn't specify whether it is relative to the ground or the water? When it mentions just the speed is that just relative to the water flowing $2ms^{-1}$ down (so upstream speed is 10, downstream is 12)? or something else entirely?

With aeroplanes and crosswind: "A jet moving initially at $485 kmh^{-1}$ due east enters region where wind is blowing $1.60 \times 10^2$ in the direction 30 degrees due north of east. Calculate the speed and direction (velocity) of the jet relative to the ground." Here, is the jet's movement initially air speed (relative to air) or relative to ground? What about wind? Also, if air speed is $x km h^{-1}$, is that x relative to the ground (so wind originally is taken into account) or is x just in the air nothing else.

Appreciate if you could answer all above. I'm really confused.

Answer 1

A useful way to remember how to work out relative velocities is as follows.

Suppose the velocity of object $a$ relative to the ground is $\vec v_{\rm ag}$ and the velocity of object $b$ relative to the ground is $\vec v_{\rm bg}$ then the following equation can be used to find the velocity of object $a$ relative to object $b$, $\vec v_{\rm ab}$.

$\vec v_{\rm a\Large \color{red}b}+ \vec v_{\rm {\Large \color{red}b}g} = \vec v_{\rm ag}$

Note the sequence of subscripts.

In general if you have a statement like, the speed of the object is $\,\dots$ with no further qualification, you take it as relative to whatever you think is a sensible choice.
Thus for a boat - still water, a walker - the ground, a wave - still medium etc.