Positions of A and B are given as functions of time 't' as `barR_A = (3t hati +4hatj) m, barR_B =(hati+athatj)m`, where a is a constant and t is time in seconds. If relative velocity of A with respect to B is `(3 hati +7hatj) ms^(-1)`, then find the value of a.

Answer

To solve the problem, we need to find the value of the constant \( a \) given the positions of points A and B as functions of time and the relative velocity of A with respect to B.

### Step-by-Step Solution:

1. **Write the position vectors of A and B**:
\[ \bar{R}_A = (3t \hat{i} + 4 \hat{j}) \, \text{m} \]
\[ \bar{R}_B = (\hat{i} + a \hat{j}) \, \text{m} \]

2. **Find the velocity of A**:
The velocity of A is the derivative of its position vector with respect to time \( t \):
\[ \bar{V}_A = \frac{d\bar{R}_A}{dt} = \frac{d}{dt}(3t \hat{i} + 4 \hat{j}) = 3 \hat{i} + 0 \hat{j} = 3 \hat{i} \, \text{m/s} \]

3. **Find the velocity of B**:
The velocity of B is the derivative of its position vector with respect to time \( t \):
\[ \bar{V}_B = \frac{d\bar{R}_B}{dt} = \frac{d}{dt}(\hat{i} + a \hat{j}) = 0 \hat{i} + a \hat{j} = a \hat{j} \, \text{m/s} \]

4. **Calculate the relative velocity of A with respect to B**:
The relative velocity of A with respect to B is given by:
\[ \bar{V}_{A/B} = \bar{V}_A - \bar{V}_B = (3 \hat{i} + 0 \hat{j}) - (0 \hat{i} + a \hat{j}) = 3 \hat{i} - a \hat{j} \]

5. **Set the relative velocity equal to the given value**:
According to the problem, the relative velocity of A with respect to B is given as:
\[ \bar{V}_{A/B} = 3 \hat{i} + 7 \hat{j} \, \text{m/s} \]
Therefore, we can set the two expressions for relative velocity equal to each other:
\[ 3 \hat{i} - a \hat{j} = 3 \hat{i} + 7 \hat{j} \]

6. **Compare the coefficients**:
From the equation above, we can compare the coefficients of \( \hat{j} \):
\[ -a = 7 \]

7. **Solve for \( a \)**:
\[ a = -7 \]

### Final Answer:
The value of \( a \) is \( -7 \).

---