Why Is the Relative Velocity Between Two Opposing Balls Added?

[mohabitar]

I'm watching a lecture video describing the following situation:

There is a red ball and a green ball on a collision course. The green ball moves with a constant velocity 6 m/s due East, while the red ball moves with a constant velocity 2 m/s due west.

They are saying that the velocity of the green ball in the frame of the red ball is 8 m/s. I'm not understanding why it's 8. Obviously they are adding the two velocities together, but why? One would think that since green is headed east and red is headed west, that we should subtract the values. What's the reasoning behind this? Thanks.

 

[diazona]

Imagine yourself moving along with the red ball. Maybe that will make it a bit clearer.

For a slightly more detailed explanation: to find relative velocity, you should actually subtract the two velocities. But velocity is a vector, which means it's not just the number of m/s that matters, it's the direction. For objects moving along a line, you indicate the direction by a sign: to the right (east) is positive, and to the left (west) is negative. So in your example, the green ball has velocity +6 m/s, but the red ball has velocity -2 m/s. When you subtract those, (green)-(red), you get +8 m/s. Thus the velocity of the green ball in the frame of the red ball is +8 m/s.

To see why you subtract instead of adding, imagine a new situation, a red ball and a green ball both moving to the east at 4 m/s. Both of them are moving in the same direction, so they both have the same sign on the velocity: both velocities are +4 m/s. You agree that the velocity of the green ball in the frame of the red ball is 0, right? Obviously adding the numbers doesn't give you the right answer. But subtracting them does.

 

[mohabitar]

Thanks a lot for your answer. Really helped clear things up. However, what exactly does 'in the reference of' or 'in the frame of' really mean?

 

[diazona]

It's kind of like "as if you were traveling along with". So when they ask for the velocity of the green ball in the frame of the red ball, they want to know what velocity you would observe the green ball to have if you were moving along with the red ball.

 

[rwisz]

The reasoning behind adding the velocities in this situation is due to the concept of relative motion. In this case, we are considering the motion of the green ball from the perspective of the red ball. From the perspective of the red ball, the green ball appears to be moving with a velocity of 6 m/s due east. However, since the red ball is also moving with a velocity of 2 m/s due west, the effective relative velocity of the green ball from the perspective of the red ball is the sum of these two velocities, which is 8 m/s due east. This is because both balls are moving in opposite directions, so their velocities are added together to determine the relative motion between them. This concept is important in understanding the motion of objects in different frames of reference and is a fundamental principle in physics. I hope this explanation helps clarify the concept for you.