If the mass of a moving object is tripled and its velocity is doubled, the kinetic energy will be multiplied by

Options

A. a. 8
B. b. 6
C. c. 2
D. d. 12

Explanation

To solve the problem, we need to understand the relationship between mass, velocity, and kinetic energy. The formula for kinetic energy (KE) is given by:

\[ KE = \frac{1}{2} mv^2 \]

where \( m \) is the mass of the object and \( v \) is its velocity.

### Step 1: Define Initial Conditions
Let:
- Initial mass \( m_1 = m \)
- Initial velocity \( v_1 = v \)

The initial kinetic energy \( KE_1 \) can be calculated as:
\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} m v^2 \]

### Step 2: Define New Conditions
According to the problem:
- The mass is tripled, so \( m_2 = 3m \)
- The velocity is doubled, so \( v_2 = 2v \)

### Step 3: Calculate Final Kinetic Energy
Now, we can calculate the final kinetic energy \( KE_2 \) using the new mass and velocity:
\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (3m) (2v)^2 \]

### Step 4: Simplify the Expression
Now we simplify \( KE_2 \):
\[ KE_2 = \frac{1}{2} (3m) (4v^2) = \frac{1}{2} \cdot 3m \cdot 4v^2 = 6mv^2 \]

### Step 5: Relate Final Kinetic Energy to Initial Kinetic Energy
Now we relate \( KE_2 \) to \( KE_1 \):
\[ KE_1 = \frac{1}{2} mv^2 \]
To find the factor by which the kinetic energy has increased, we can divide \( KE_2 \) by \( KE_1 \):
\[ \text{Factor} = \frac{KE_2}{KE_1} = \frac{6mv^2}{\frac{1}{2} mv^2} \]

### Step 6: Calculate the Factor
Now, simplifying the fraction:
\[ \text{Factor} = \frac{6mv^2}{\frac{1}{2} mv^2} = \frac{6}{\frac{1}{2}} = 6 \times 2 = 12 \]

### Conclusion
Thus, the kinetic energy will be multiplied by **12**.

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