A body of mass 0.02 kg falls from a height of 5 metre into a pile of sand. The body penetrates the sand a distance of 5 cm before stoping. What force has the sand exerted on the body ?

Options

A. `1.96 N`
B. `-19.6N`
C. `-0.196N`
D. `0.0196N`

Answer

To find the force exerted by the sand on the body, we can follow these steps:

### Step 1: Calculate the velocity of the body just before it hits the sand.
We can use the equation of motion:
\[ v^2 = u^2 + 2as \]
where:
- \( v \) = final velocity (just before hitting the sand)
- \( u \) = initial velocity (0 m/s, since it starts from rest)
- \( a \) = acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))
- \( s \) = height from which the body falls (5 m)

Substituting the values:
\[ v^2 = 0 + 2 \times 9.8 \times 5 \]
\[ v^2 = 98 \]
\[ v = \sqrt{98} \approx 9.9 \, \text{m/s} \]

### Step 2: Calculate the change in kinetic energy as the body penetrates the sand.
The kinetic energy (KE) just before hitting the sand is given by:
\[ KE = \frac{1}{2} mv^2 \]
where:
- \( m \) = mass of the body (0.02 kg)
- \( v \) = velocity just before hitting the sand (9.9 m/s)

Substituting the values:
\[ KE = \frac{1}{2} \times 0.02 \times (9.9)^2 \]
\[ KE = \frac{1}{2} \times 0.02 \times 98 \]
\[ KE = 0.01 \times 98 = 0.98 \, \text{J} \]

### Step 3: Calculate the work done by the sand.
The work done by the sand (W) is equal to the change in kinetic energy, which is equal to the initial kinetic energy since the final kinetic energy is 0 (the body comes to rest):
\[ W = -KE = -0.98 \, \text{J} \]

### Step 4: Relate work done to force exerted by the sand.
The work done by the force exerted by the sand can also be expressed as:
\[ W = F \times d \]
where:
- \( F \) = force exerted by the sand
- \( d \) = distance penetrated into the sand (5 cm = 0.05 m)

Rearranging gives:
\[ F = \frac{W}{d} \]
Substituting the values:
\[ F = \frac{-0.98}{0.05} \]
\[ F = -19.6 \, \text{N} \]

### Conclusion:
The force exerted by the sand on the body is approximately \( -19.6 \, \text{N} \). The negative sign indicates that the force exerted by the sand is in the opposite direction to the motion of the body.

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