A strip of wood of mass `M` and length `l` is placed on a smooth horizontal surface. An insect of mass `m` starts at one end of the strip and walks to the other end in time `t`, moving with a constant speed. The speed of the insect as seen from the ground is

Options

A. `l/t(M/(M+m))`
B. `l/t(m/(M+m))`
C. `l/t(M/m)`
D. `l/t(m/M)`

Explanation

To solve the problem, we need to find the speed of the insect as seen from the ground. Let's break it down step by step.

### Step 1: Understand the system
We have a strip of wood with mass \( M \) and length \( l \) on a smooth horizontal surface. An insect with mass \( m \) walks from one end of the strip to the other in time \( t \) with a constant speed.

### Step 2: Define variables
Let:
- \( u \) = speed of the insect as seen from the ground
- \( v \) = speed of the strip of wood as it moves in the opposite direction to the insect

### Step 3: Initial momentum
Initially, the system (insect + strip) is at rest, so the total initial momentum is zero:
\[ 0 = M \cdot 0 + m \cdot 0 \]

### Step 4: Final momentum
As the insect walks to the other end of the strip, the wood will move in the opposite direction to conserve momentum. The final momentum of the system can be expressed as:
\[ m \cdot u + M \cdot (-v) = 0 \]
This simplifies to:
\[ m \cdot u = M \cdot v \quad \text{(1)} \]

### Step 5: Relate speeds in the ground frame
In the ground frame, the speed of the insect as it walks on the strip is:
\[ v' = u + v \]
where \( v' \) is the speed of the insect relative to the ground.

### Step 6: Distance traveled by the insect
The insect travels the length \( l \) of the strip in time \( t \). Therefore, we can express the relationship as:
\[ v' \cdot t = l \quad \text{(2)} \]

### Step 7: Substitute \( v' \) from (2) into (1)
From equation (2), we have:
\[ (u + v) \cdot t = l \]
This can be rearranged to:
\[ u + v = \frac{l}{t} \quad \text{(3)} \]

### Step 8: Solve for \( v \)
From equation (1), we can express \( v \) in terms of \( u \):
\[ v = \frac{m}{M} u \quad \text{(4)} \]
Substituting equation (4) into equation (3):
\[ u + \frac{m}{M} u = \frac{l}{t} \]
Factoring out \( u \):
\[ u \left(1 + \frac{m}{M}\right) = \frac{l}{t} \]
Thus,
\[ u = \frac{l}{t} \cdot \frac{M}{M + m} \quad \text{(5)} \]

### Step 9: Final expression for speed
The speed of the insect as seen from the ground is:
\[ u = \frac{l}{t} \cdot \frac{M}{M + m} \]

### Summary
The speed of the insect as seen from the ground is given by:
\[ u = \frac{l}{t} \cdot \frac{M}{M + m} \]