The velocity of an object moving rectilinearly is given as a function of time by `v=4t-3t^(2)` where `v` is in m/s and `t` is in seconds. The average velocity 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 \( t = 0 \) and \( t = 2 \) seconds, we can follow these steps:

### Step 1: Understand the formula for average velocity
The average velocity \( V_{avg} \) over a time interval is given by the formula:
\[ V_{avg} = \frac{\Delta s}{\Delta t} \]
where \( \Delta s \) is the total displacement and \( \Delta t \) is the total time.

### Step 2: Calculate total displacement \( \Delta s \)
The total displacement can be found by integrating the velocity function \( v(t) = 4t - 3t^2 \) over the interval from \( t = 0 \) to \( t = 2 \):
\[ \Delta s = \int_{0}^{2} v(t) \, dt = \int_{0}^{2} (4t - 3t^2) \, dt \]

### Step 3: Perform the integration
Now, we will calculate the integral:
\[ \Delta s = \int_{0}^{2} (4t - 3t^2) \, dt = \left[ 2t^2 - t^3 \right]_{0}^{2} \]

### Step 4: Evaluate the integral at the limits
Now, we evaluate the expression at the limits:
\[ \Delta s = \left[ 2(2)^2 - (2)^3 \right] - \left[ 2(0)^2 - (0)^3 \right] \]
Calculating this gives:
\[ \Delta s = \left[ 2 \times 4 - 8 \right] - [0] = 8 - 8 = 0 \]

### Step 5: Calculate total time \( \Delta t \)
The total time \( \Delta t \) from \( t = 0 \) to \( t = 2 \) seconds is:
\[ \Delta t = 2 - 0 = 2 \text{ seconds} \]

### Step 6: Calculate average velocity
Now, substituting the values of \( \Delta s \) and \( \Delta t \) into the average velocity formula:
\[ V_{avg} = \frac{\Delta s}{\Delta t} = \frac{0}{2} = 0 \text{ m/s} \]

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