A particle moves on x-axis such that its KE varies as a relation KE=`3t^(2)` then the average kinetic energy of a particle in 0 to 2 sec is given by :

Options

A. 6 joule
B. 8 joule
C. 4 joule
D. 16 joule

Answer

To find the average kinetic energy of a particle moving along the x-axis, where the kinetic energy (KE) varies as \( KE = 3t^2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Function for Kinetic Energy**:
The kinetic energy is given by the equation:
\[ KE(t) = 3t^2 \]

2. **Set Up the Formula for Average Value**:
The average value of a function \( f(t) \) over an interval \([a, b]\) is given by:
\[ \text{Average} = \frac{\int_a^b f(t) \, dt}{\int_a^b \, dt} \]
In this case, \( f(t) = 3t^2 \), \( a = 0 \), and \( b = 2 \).

3. **Calculate the Integral of the Kinetic Energy**:
We need to compute the integral of \( 3t^2 \) from 0 to 2:
\[ \int_0^2 3t^2 \, dt \]
To solve this, we find the antiderivative:
\[ \int 3t^2 \, dt = 3 \cdot \frac{t^3}{3} = t^3 \]
Now, evaluate it from 0 to 2:
\[ \left[ t^3 \right]_0^2 = 2^3 - 0^3 = 8 - 0 = 8 \]

4. **Calculate the Integral of dt**:
Next, we compute the integral of \( dt \) from 0 to 2:
\[ \int_0^2 dt = [t]_0^2 = 2 - 0 = 2 \]

5. **Calculate the Average Kinetic Energy**:
Now, we can find the average kinetic energy:
\[ \text{Average KE} = \frac{\int_0^2 3t^2 \, dt}{\int_0^2 dt} = \frac{8}{2} = 4 \]

6. **Final Answer**:
Thus, the average kinetic energy of the particle from 0 to 2 seconds is:
\[ \text{Average KE} = 4 \text{ joules} \]