A graph is plotted by taking transnational kinetic energy of body as ordinate and velocity as abscissa. Slope of the graph at an instant represents its

A. Velocity
B. Mass
C. Momentum
D. Kinetic energy pre unit mass

Options

A. Velocity
B. Mass
C. Momentum
D. Kinetic energy per unit mass

Answer

To solve the question step by step, we will analyze the relationship between translational kinetic energy, velocity, and the slope of the graph.

### Step 1: Understand the Variables
- The graph is plotted with translational kinetic energy (KE) on the y-axis and velocity (v) on the x-axis.
- The formula for translational kinetic energy is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the body and \( v \) is its velocity.

### Step 2: Identify the Shape of the Graph
- Since the kinetic energy is proportional to the square of the velocity, the graph of kinetic energy versus velocity will be a parabolic curve.

### Step 3: Determine the Slope of the Graph
- The slope of a graph at any point is defined as:
\[ \text{slope} = \frac{\Delta y}{\Delta x} \]
In this case, it would be:
\[ \text{slope} = \frac{\Delta KE}{\Delta v} \]

### Step 4: Differentiate the Kinetic Energy Equation
- To find the slope, we can differentiate the kinetic energy equation with respect to velocity:
\[ KE = \frac{1}{2} mv^2 \]
Taking the derivative with respect to \( v \):
\[ \frac{d(KE)}{dv} = mv \]
This indicates that the slope of the graph at any point is equal to \( mv \).

### Step 5: Relate the Slope to Momentum
- The momentum \( p \) of the body is defined as:
\[ p = mv \]
- Therefore, the slope of the graph, which is \( mv \), represents the momentum of the body.

### Conclusion
- The slope of the graph at an instant represents the momentum of the body. Thus, the correct answer is:
**C. Momentum**