The arrow diagram

To calculate the velocity of an object relative to a different reference frame, we need the Galilean composition of velocities. Instead of memorising formulas with subscripts to flip around in your head, it is convenient to use an arrow diagram: each object (or reference frame) is a node, each arrow is a known velocity, read as “velocity of the starting object as seen from the arrival object”.

The notation is the key: $\vec{v}_{X(Y)}$ means “velocity of $X$ relative to $Y$”, i.e. who is moving is outside the brackets, who is watching is inside.

Principle — Arrow diagram rule

To find the velocity of an object $A$ relative to an observer $B$:

1. Draw a node for each object/reference frame involved.
2. Draw an arrow from $X$ to $Y$ labelled with the velocity $\vec{v}_{X(Y)}$ (velocity of $X$ relative to $Y$), if we know it.
3. Follow a path from $A$ to $B$ across the known arrows. For every arrow crossed in the opposite direction, flip the sign of the velocity.
4. The algebraic sum of the velocities along the path gives the required velocity $\vec{v}_{A(B)}$.

The advantage is that there is no need to remember which formula to use: just follow the path on the graph and the signs sort themselves out. Crossing an arrow “forwards” adds its velocity; crossing it “backwards” subtracts it.

Arrow diagram for the Galilean composition: each arrow is a known velocity. To find an unknown velocity, compose a path across the arrows; arrows crossed backwards flip sign.

The worked exercise Il gatto sul carrello applies the arrow diagram to a complete case, including the switch to the centre-of-mass reference frame.

Links

Topics: Centro di massa Skills: Cambio di sistema di riferimento

Related exercises: Esercizio svolto — Il gatto sul carrello · Razzo che esplode a metà traiettoria · Dimostrazione del teorema di König