A body is moving with variable acceleartion (a) along a straight line. The average acceleration of body in time interval `t_(1)" to "t_(2)` is :-

Options

A. `(a[t_(2)+t_(1)])/(2)`
B. `(a[t_(2)-t_(1)])/(2)`
C. `(int_(t_(1))^(t_(2))adt)/(t_(2)+t_(1))`
D. `(int_(t_(1))^(t_(2))adt)/(t_(2)-t_(1))`

Answer

To find the average acceleration of a body moving with variable acceleration \( a(t) \) over a time interval from \( t_1 \) to \( t_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Average Acceleration**:
The average acceleration \( \bar{a} \) over a time interval is defined as the change in velocity divided by the time interval. For variable acceleration, we can express this in terms of integration.

2. **Formula for Average Acceleration**:
The average acceleration can be calculated using the formula:
\[ \bar{a} = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} a(t) \, dt \]
where \( a(t) \) is the acceleration as a function of time.

3. **Setting Up the Integral**:
We need to evaluate the integral of acceleration \( a(t) \) from \( t_1 \) to \( t_2 \):
\[ \int_{t_1}^{t_2} a(t) \, dt \]

4. **Calculating the Time Interval**:
The time interval \( t_2 - t_1 \) is simply the difference between the upper and lower limits of the time.

5. **Putting It All Together**:
Now, substituting the integral and the time interval into the average acceleration formula:
\[ \bar{a} = \frac{\int_{t_1}^{t_2} a(t) \, dt}{t_2 - t_1} \]

6. **Conclusion**:
Thus, the average acceleration of the body in the time interval from \( t_1 \) to \( t_2 \) is given by:
\[ \bar{a} = \frac{\int_{t_1}^{t_2} a(t) \, dt}{t_2 - t_1} \]