A monkey is climbing up a tree at a speed of 3 m/s. A dog runs towards the tree with a speed of 4 m/s. What is the relative speed of the dog as seen by the monkey?

Options

A. `gt7m//s`
B. Between 5m/s and 7m/s
C. 5m/sec
D. `lt 5m//s`

Answer

To find the relative speed of the dog as seen by the monkey, we can follow these steps:

### Step 1: Identify the Speeds
- The monkey is climbing up the tree at a speed of \( v_m = 3 \, \text{m/s} \).
- The dog is running towards the tree at a speed of \( v_d = 4 \, \text{m/s} \).

### Step 2: Understand the Direction of Motion
- The monkey is moving vertically upwards.
- The dog is moving horizontally towards the tree.

### Step 3: Use the Relative Velocity Formula
The relative velocity of the dog with respect to the monkey can be calculated using the formula:
\[ v_{dm} = v_d - v_m \]
However, since the directions are perpendicular (the monkey is moving up and the dog is moving towards the tree), we need to use the Pythagorean theorem to find the resultant speed.

### Step 4: Calculate the Resultant Speed
Since the velocities are perpendicular, we can calculate the magnitude of the relative velocity using:
\[ v_{dm} = \sqrt{v_d^2 + v_m^2} \]
Substituting the values:
\[ v_{dm} = \sqrt{(4 \, \text{m/s})^2 + (3 \, \text{m/s})^2} \]
\[ = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{m/s} \]

### Step 5: Conclusion
The relative speed of the dog as seen by the monkey is \( 5 \, \text{m/s} \).

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