derive the formula for the kinetic energy of an object of mass m moving with velocity v [class 9 solution]

To derive the formula for the kinetic energy of an object of mass \( m \) moving with velocity \( v \), we can follow these steps:

### Step 1: Understand the Concept of Work
Kinetic energy is the energy possessed by an object due to its motion. According to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. Work is defined as the force applied to an object multiplied by the distance over which the force is applied.

### Step 2: Express Work in Terms of Force and Distance
The work \( W \) done by a force \( F \) over a distance \( d \) is given by:
\[
W = F \cdot d
\]

### Step 3: Relate Force to Mass and Acceleration
According to Newton's second law of motion, force \( F \) is related to mass \( m \) and acceleration \( a \) by:
\[
F = m \cdot a
\]

### Step 4: Express Distance in Terms of Initial and Final Velocity
If the object starts from rest (initial velocity \( u = 0 \)) and accelerates to a final velocity \( v \), the distance \( d \) traveled can be found using the equation of motion:
\[
v^2 = u^2 + 2ad
\]
Since \( u = 0 \), this simplifies to:
\[
v^2 = 2ad
\]
So, the distance \( d \) can be expressed as:
\[
d = \frac{v^2}{2a}
\]

### Step 5: Substitute into the Work Formula
Substitute the expressions for force and distance into the work formula:
\[
W = F \cdot d = m \cdot a \cdot \frac{v^2}{2a}
\]
The acceleration \( a \) cancels out, leaving:
\[
W = \frac{1}{2} m v^2
\]

### Step 6: Interpret the Result as Kinetic Energy
The work done on the object is equal to the kinetic energy \( K \) of the object. Therefore:
\[
\text{Kinetic Energy}, K = \frac{1}{2} m v^2
\]

This is the formula for the kinetic energy of an object of mass \( m \) moving with velocity \( v \).

### Summary
The kinetic energy \( K \) of an object is given by:
\[
K = \frac{1}{2} m v^2
\]
where:
- \( m \) is the mass of the object.
- \( v \) is the velocity of the object.

Do you want more details or have any questions?

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**Related Questions:**

1. What is the difference between kinetic energy and potential energy?
2. How does kinetic energy change if the velocity of an object doubles?
3. Can kinetic energy ever be negative?
4. How is kinetic energy related to momentum?
5. What are some examples of kinetic energy in everyday life?
6. How does mass affect the kinetic energy of an object?
7. What is the work-energy theorem and how does it relate to kinetic energy?
8. How would you derive the formula for potential energy?

**Tip:** Remember, kinetic energy depends on the square of the velocity, so even small changes in speed can significantly affect the energy of a moving object!

Learn how to derive the formula for kinetic energy of an object of mass m moving with velocity v with a step-by-step explanation. Understand the concepts of work, force, and Newton's laws of motion. Suitable for Class 9 students studying physics.

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