A packet of weight w is dropped with the help of a parachute and on striking the ground comes to rest with retardation equal to twice the acceleration due to gravity. The force exerted on the ground is

Options

A. w
B. 2w
C. 3w
D. 4w

Answer

To solve the problem step by step, we need to analyze the forces acting on the packet when it strikes the ground and comes to rest.

### Step 1: Understand the forces acting on the packet
When the packet is dropped, it has a weight \( W \) acting downward. The weight is given by the equation:
\[ W = mg \]
where \( m \) is the mass of the packet and \( g \) is the acceleration due to gravity.

### Step 2: Identify the retardation
The problem states that the packet comes to rest with a retardation equal to twice the acceleration due to gravity. Therefore, the retardation \( a \) can be expressed as:
\[ a = 2g \]

### Step 3: Apply Newton's second law
According to Newton's second law, the net force acting on the packet when it strikes the ground can be expressed as:
\[ F_{\text{net}} = ma \]
where \( F_{\text{net}} \) is the net force acting on the packet, and \( a \) is the retardation.

### Step 4: Calculate the mass of the packet
From the weight equation \( W = mg \), we can express the mass \( m \) as:
\[ m = \frac{W}{g} \]

### Step 5: Set up the equation for net force
When the packet hits the ground, the forces acting on it are:
- The weight \( W \) acting downward.
- The force \( F \) exerted by the ground acting upward.

The net force can be expressed as:
\[ F_{\text{net}} = F - W \]

### Step 6: Substitute into Newton's second law
Now we can substitute the expressions we have into Newton's second law:
\[ F - W = ma \]
Substituting \( m \) and \( a \):
\[ F - W = \left(\frac{W}{g}\right)(2g) \]

### Step 7: Simplify the equation
This simplifies to:
\[ F - W = 2W \]
Now, we can solve for \( F \):
\[ F = 2W + W \]
\[ F = 3W \]

### Step 8: Conclusion
The force exerted on the ground is:
\[ F = 3W \]