Relative Velocity - Mistake in Textbook?

[HenryA.]

I am just reading through this free online textbook and it seems to me that there is a mistake on page 24.

http://www.anselm.edu/internet/physics/cbphysics/downloadsI/cbPhysicsIa18.pdf#page=23

He describes the muzzle velocity, which he defines as the relative velocity between two objects, as being the sum of the two objects. Which in this case is:

v_M=v_B+v_C

This is not the relative velocity, this would be the relative velocity between these two objects:

v_M=v_B-v_C

Am I missing something really simple or is this actually a mistake?

 

[HallsofIvy]

You would be correct if $v_B$ and $v_C$ were "vectors" or (in one dimension) "signed quantities" so that with the cannon ball going to the right, $v_B$ were positive and with the cannon rolling to the left, $v_C$ would be negative. But here they are clearly using the speeds or "unsigned quantities", not velocities.

For example, if the cannon ball went to the right at, say, 300 m/s while the cannon rolled back at 2 m/s. then, as velocities or "signed quantities" we would say that $v_B= 300$ and $v_C= -2$ so that the "relative velocity" would be $v_B- v_C= 300- (-2)= 302$ m/s. But the book is using the "unsigned" speeds: $v_B= 300$ m/s to the right and $v_C= 2$ to the left so that the relative speed is $v_B+ v_C= 300+ 2= 302$ m/s to the right.

 

[HenryA.]

But here they are clearly using the speeds or "unsigned quantities", not velocities.

Yes, this would make sense. I guess their use of the word velocity threw me off.

 

[Redbelly98]

I guess their use of the word velocity threw me off.

That's understandable. Reading it carefully, I noticed they make statements along the lines of "the velocity is v_B to the right". Since they specify a direction, they are correct to use "velocity" rather than "speed". However, v_B by itself (with no direction specified) is a speed, not a velocity.