Two bodies of different masses are dropped similtaneously from the top of a tower .If air resistance is proportional to the mass of the body.

Options

A. the heavier body reaches the ground earlier
B. the lighter body reaches the ground earlier
C. both reach the ground simultaneously
D. cannot be decided

Answer

To solve the problem of two bodies of different masses being dropped simultaneously from the top of a tower, we need to analyze the forces acting on each body and how these forces affect their motion. Here’s a step-by-step solution:

### Step 1: Identify the Forces Acting on the Bodies
When a body is dropped, two main forces act on it:
- The gravitational force (weight) acting downwards, which is given by \( F_g = mg \), where \( m \) is the mass of the body and \( g \) is the acceleration due to gravity.
- The air resistance acting upwards, which is proportional to the mass of the body. We can express this as \( F_{air} = m \alpha \), where \( \alpha \) is a constant of proportionality.

**Hint:** Remember that the gravitational force increases with mass, but so does the air resistance.

### Step 2: Write the Equation of Motion
According to Newton's second law, the net force acting on the body can be expressed as:
\[ F_{net} = F_g - F_{air} \]
Substituting the expressions for \( F_g \) and \( F_{air} \):
\[ F_{net} = mg - m\alpha \]

### Step 3: Apply Newton's Second Law
Using Newton's second law, we can relate the net force to the acceleration of the body:
\[ ma = mg - m\alpha \]
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ a = g - \alpha \]

**Hint:** This equation shows that the acceleration \( a \) is independent of the mass \( m \).

### Step 4: Analyze the Result
From the equation \( a = g - \alpha \), we see that the acceleration \( a \) does not depend on the mass of the body. This means that both bodies, regardless of their different masses, will experience the same acceleration as they fall.

### Step 5: Conclusion
Since both bodies have the same acceleration and are dropped simultaneously from the same height, they will reach the ground at the same time.

**Final Answer:** Both bodies will reach the ground simultaneously.