The ends of a rod AB which is 5 m long moves along two grooves `OX, OY` which are at right angles. If A moves 2 at a constant speed of `1/2m/s`, what is the speed of B, when it is 4 m from O?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry
We have a rod AB of length 5 m, where point A moves along the x-axis (OX) and point B moves along the y-axis (OY). The coordinates of point A can be represented as (x, 0) and the coordinates of point B as (0, y). Since the rod is of fixed length, we can use the Pythagorean theorem to relate x and y.

### Step 2: Set Up the Equation
Using the Pythagorean theorem, we have:
\[ x^2 + y^2 = 5^2 \]
This simplifies to:
\[ x^2 + y^2 = 25 \]

### Step 3: Differentiate with Respect to Time
To find the relationship between the rates of change of x and y, we differentiate both sides of the equation with respect to time \( t \):
\[ \frac{d}{dt}(x^2) + \frac{d}{dt}(y^2) = \frac{d}{dt}(25) \]
Using the chain rule, we get:
\[ 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \]
Dividing through by 2 gives:
\[ x \frac{dx}{dt} + y \frac{dy}{dt} = 0 \]

### Step 4: Substitute Known Values
We know:
- The speed of point A, \( \frac{dx}{dt} = v_A = \frac{1}{2} \) m/s (moving away from O).
- We need to find \( \frac{dy}{dt} = v_B \) when point B is 4 m from O, which means \( y = 4 \).

From the Pythagorean theorem, when \( y = 4 \):
\[ x^2 + 4^2 = 25 \implies x^2 + 16 = 25 \implies x^2 = 9 \implies x = 3 \text{ m} \]

### Step 5: Plug Values into the Differentiated Equation
Substituting \( x = 3 \), \( y = 4 \), and \( \frac{dx}{dt} = \frac{1}{2} \) into the differentiated equation:
\[ 3 \left(\frac{1}{2}\right) + 4 \frac{dy}{dt} = 0 \]
This simplifies to:
\[ \frac{3}{2} + 4v_B = 0 \]

### Step 6: Solve for \( v_B \)
Rearranging gives:
\[ 4v_B = -\frac{3}{2} \implies v_B = -\frac{3}{8} \text{ m/s} \]

### Conclusion
The speed of point B when it is 4 m from O is:
\[ v_B = -\frac{3}{8} \text{ m/s} \]
The negative sign indicates that point B is moving downward along the y-axis.