Calculating relative velocity: What am I doing wrong? [closed]

Question

There are two objects A and B. Points P1, P2, P3 are in the same line, and P2 is exactly at the middle of P1 and P3

Suppose B is moving at constant velocity along the line P1 to P3. Thus, time taken by B to reach from P1 to P2 is same as that from P2 to P3. Let's assume this time is 1 second.

But if we observe B's velocity relative to A

'A' measured it's distance from 'B' when 'B' was at P1.

After 1 second 'A' again measured it's distance from 'B' when 'B' was at P2. So 'A' will see relative distance D1 travelled by 'B'

D1 = Distance (A to P2) - Distance (A to P1)

After 1 second it again measured it's distance from 'B' when 'B' was at P3. So 'A' will see relative distance D2 travelled by 'B'

D2 = Distance (A to P3) - Distance (A to P2)

Now, by geometry we can see D2 > D1

That implies in same amount of time B travelled more distance.

Hence, 'A' will see 'B' is accelerating, which is not correct. What mistake am I doing in calculating relative velocity?

Note: This is not homework question. When objects are colinear the approach works as expected and I wanted to know how do they calculate relative velocity in this situation. The question didn't ask how to solve something as it already discuss approach taken. There are many "What am I doing wrong" questions on this forum. Hence I thought to ask this one.

Answer 1

The mistake is in making differences between positions as they were scalars, while they are vectors.

Consider the vectors $\vec{r}_1$, $\vec{r}_2$, and $\vec{r}_3$ corresponding to the position of $P_1$, $P_2$, and $P_3$ from $A$, respectively. The magnitude of these vectors are the distances of $P_1$, $P_2$, and $P_3$ from $A$. Now, if you just compute the differences between these magnitudes, you get the results in your questions, which are not correct. In fact, you should compute the differences between vectors:

• $\vec{r}_2 - \vec{r}_1$ is the position vector of $P_2$ with respect to $P_1$.
• $\vec{r}_3 - \vec{r}_2$ is the position vector of $P_3$ with respect to $P_2$.

Computing the magnitude of these differences gives you the right result.

Another way to understand the same mistake is the following: take a triangle with side length $a$, $b$, and $c$. For a generic triangle, $c \neq a + b$. However, in your post you are saying that $c = a + b$, which is indeed incorrect.