A body falls from height h. The `v-s` graph is :

Options

A.
B.
C.
D.

Answer

To solve the problem of determining the correct v-s (velocity-displacement) graph for a body falling from a height \( h \), we can follow these steps:

### Step 1: Understand the motion of the body
When a body falls freely under the influence of gravity, its motion can be described using the equations of motion. The initial velocity \( u \) is 0 (since it starts from rest), and the acceleration \( a \) is equal to \( g \) (acceleration due to gravity).

### Step 2: Apply the equation of motion
We can use the equation of motion:
\[ v^2 = u^2 + 2as \]
Substituting \( u = 0 \) and \( a = g \):
\[ v^2 = 0 + 2gs \]
This simplifies to:
\[ v^2 = 2gs \]

### Step 3: Differentiate the equation
To find the relationship between \( v \) and \( s \), we differentiate \( v^2 = 2gs \) with respect to \( s \):
\[ \frac{d(v^2)}{ds} = 2g \]
Using the chain rule, we have:
\[ 2v \frac{dv}{ds} = 2g \]
Dividing both sides by 2:
\[ v \frac{dv}{ds} = g \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ \frac{dv}{ds} = \frac{g}{v} \]

### Step 5: Analyze the relationship
This equation shows that the change in velocity with respect to displacement \( \frac{dv}{ds} \) is inversely proportional to the velocity \( v \). This means that as the velocity increases, \( \frac{dv}{ds} \) decreases.

### Step 6: Determine the shape of the graph
Since \( \frac{dv}{ds} \) is inversely proportional to \( v \), the slope of the v-s graph decreases as \( v \) increases. Therefore, the correct v-s graph will be one that has a decreasing slope.

### Conclusion
From the options provided, the graph that shows a decreasing slope as velocity increases is the correct answer. Thus, the correct option is option 3.

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