The slope of the kinetic energy versus position vector curve gives the rate of change of :-

Options

A. work
B. force
C. power
D. momentum

Answer

To solve the question, we need to determine what the slope of the kinetic energy (KE) versus position (x) curve represents. Let's break this down step by step:

### Step 1: Understand the relationship between kinetic energy and position
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the object and \( v \) is its velocity.

### Step 2: Identify the slope of the KE vs. position graph
The slope of the kinetic energy versus position curve is defined as:
\[ \text{slope} = \frac{d(KE)}{dx} \]
This means we are looking for the rate of change of kinetic energy with respect to position.

### Step 3: Apply the chain rule
To express \( \frac{d(KE)}{dx} \) in terms of velocity, we can use the chain rule. Since \( KE \) depends on \( v \), and \( v \) depends on \( x \), we can write:
\[ \frac{d(KE)}{dx} = \frac{d(KE)}{dv} \cdot \frac{dv}{dx} \]

### Step 4: Differentiate kinetic energy with respect to velocity
Using the formula for kinetic energy:
\[ \frac{d(KE)}{dv} = m v \]
So, we can substitute this into our equation:
\[ \frac{d(KE)}{dx} = mv \cdot \frac{dv}{dx} \]

### Step 5: Relate \( \frac{dv}{dx} \) to acceleration
From kinematics, we know that:
\[ a = \frac{dv}{dt} \]
and
\[ v = \frac{dx}{dt} \]
Thus, we can express \( \frac{dv}{dx} \) as:
\[ \frac{dv}{dx} = \frac{a}{v} \]

### Step 6: Substitute back into the equation
Substituting \( \frac{dv}{dx} \) back into our equation gives us:
\[ \frac{d(KE)}{dx} = mv \cdot \frac{a}{v} = ma \]
This shows that the slope of the kinetic energy versus position curve is equal to the force acting on the object.

### Conclusion
Thus, the slope of the kinetic energy versus position vector curve gives the rate of change of **momentum**.