The velocity of an object moving rectillinearly is given as a functiion of time by `v=4t-3t^(2)` where `v` is in m/s and `t` is in seconds. The average velociy if particle between `t=0` to `t=2` seconds is

Options

A. 0
B. `-2m//s`
C. `-4m//s`
D. `+2m//s`

Answer

To find the average velocity of the particle between \( t = 0 \) seconds and \( t = 2 \) seconds, we can follow these steps:

### Step 1: Understand the average velocity formula
The average velocity \( V_{avg} \) over a time interval can be calculated using the formula:
\[ V_{avg} = \frac{\text{Displacement}}{\text{Time interval}} \]
Where displacement is the change in position of the object during the time interval.

### Step 2: Set up the velocity function
The velocity of the object is given by:
\[ v(t) = 4t - 3t^2 \]

### Step 3: Calculate the displacement
To find the displacement, we need to integrate the velocity function over the time interval from \( t = 0 \) to \( t = 2 \):
\[ s = \int_{0}^{2} v(t) \, dt = \int_{0}^{2} (4t - 3t^2) \, dt \]

### Step 4: Perform the integration
Now we can compute the integral:
\[ s = \int (4t - 3t^2) \, dt = 2t^2 - t^3 + C \]
Now we will evaluate this from \( t = 0 \) to \( t = 2 \):
\[ s = \left[ 2(2^2) - (2^3) \right] - \left[ 2(0^2) - (0^3) \right] \]
Calculating the first part:
\[ s = \left[ 2(4) - 8 \right] - [0] = 8 - 8 = 0 \]

### Step 5: Calculate the average velocity
Now that we have the displacement \( s = 0 \), we can calculate the average velocity:
\[ V_{avg} = \frac{s}{\Delta t} = \frac{0}{2 - 0} = 0 \]

### Conclusion
Thus, the average velocity of the particle between \( t = 0 \) seconds and \( t = 2 \) seconds is:
\[ \boxed{0 \, \text{m/s}} \]