Velocity of three particles A, B and C varies with time 't' as `barV_A = (3t hati +4hatj) ms^(-1), barV_B =(hati+2hatj)ms^(-1), barVc = (6thati +5thatj)ms^(-1)` . Then find i) acceleration of B ii) relative acceleration of A with respect to C.

Answer

To solve the problem, we need to find the acceleration of particle B and the relative acceleration of particle A with respect to particle C. Let's break it down step by step.

### Step 1: Find the acceleration of particle B

The velocity of particle B is given as:
\[ \bar{V}_B = \hat{i} + 2\hat{j} \, \text{m/s} \]
Since this velocity is constant (it does not depend on time \(t\)), the acceleration \( \bar{a}_B \) can be calculated as the derivative of the velocity with respect to time.

\[ \bar{a}_B = \frac{d}{dt} \bar{V}_B = \frac{d}{dt} (\hat{i} + 2\hat{j}) = 0 \]

Thus, the acceleration of particle B is:
\[ \bar{a}_B = 0 \, \text{m/s}^2 \]

### Step 2: Find the acceleration of particle A

The velocity of particle A is given as:
\[ \bar{V}_A = (3t \hat{i} + 4 \hat{j}) \, \text{m/s} \]
To find the acceleration of A, we take the derivative of \( \bar{V}_A \) with respect to time \(t\):

\[ \bar{a}_A = \frac{d}{dt} \bar{V}_A = \frac{d}{dt} (3t \hat{i} + 4 \hat{j}) = 3 \hat{i} + 0 = 3 \hat{i} \, \text{m/s}^2 \]

### Step 3: Find the acceleration of particle C

The velocity of particle C is given as:
\[ \bar{V}_C = (6t \hat{i} + 5 \hat{j}) \, \text{m/s} \]
To find the acceleration of C, we take the derivative of \( \bar{V}_C \) with respect to time \(t\):

\[ \bar{a}_C = \frac{d}{dt} \bar{V}_C = \frac{d}{dt} (6t \hat{i} + 5 \hat{j}) = 6 \hat{i} + 0 = 6 \hat{i} \, \text{m/s}^2 \]

### Step 4: Find the relative acceleration of A with respect to C

The relative acceleration of A with respect to C is given by:
\[ \bar{a}_{A/C} = \bar{a}_A - \bar{a}_C \]
Substituting the values we found:
\[ \bar{a}_{A/C} = (3 \hat{i}) - (6 \hat{i}) = -3 \hat{i} \, \text{m/s}^2 \]

### Final Answers
1. Acceleration of B: \( \bar{a}_B = 0 \, \text{m/s}^2 \)
2. Relative acceleration of A with respect to C: \( \bar{a}_{A/C} = -3 \hat{i} \, \text{m/s}^2 \)