A particle move with velocity `v_(1)` for time `t_(1) and v_(2)` for time `t_(2)` along a straight line. The magntidue of its average acceleration is

Options

A. `(v_(2)-v_(1))/(t_(1)-t_(2))`
B. `(v_(2)-v_(1))/(t_(1)+t_(2))`
C. `(v_(2)-t_(1))/(t_(2)-t_(1))`
D. `(v_(1)+v_(2))/(t_(1)-t_(2))`

Answer

To find the magnitude of the average acceleration of a particle that moves with different velocities over different time intervals, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Average Acceleration**:
Average acceleration is defined as the total change in velocity divided by the total time taken. Mathematically, it can be expressed as:
\[ a_{avg} = \frac{\Delta V}{\Delta t} \]
where \(\Delta V\) is the change in velocity and \(\Delta t\) is the total time.

2. **Identify the Given Values**:
- Initial velocity \(v_1\) during time \(t_1\).
- Final velocity \(v_2\) during time \(t_2\).

3. **Calculate the Change in Velocity**:
The change in velocity (\(\Delta V\)) can be calculated as:
\[ \Delta V = v_2 - v_1 \]

4. **Calculate the Total Time**:
The total time (\(\Delta t\)) during which the particle is in motion is the sum of the two time intervals:
\[ \Delta t = t_1 + t_2 \]

5. **Substitute the Values into the Average Acceleration Formula**:
Now substitute \(\Delta V\) and \(\Delta t\) into the average acceleration formula:
\[ a_{avg} = \frac{v_2 - v_1}{t_1 + t_2} \]

6. **Determine the Magnitude of Average Acceleration**:
Since acceleration is a vector quantity, we take the magnitude:
\[ |a_{avg}| = \left| \frac{v_2 - v_1}{t_1 + t_2} \right| \]

### Final Answer:
The magnitude of the average acceleration is:
\[ |a_{avg}| = \left| \frac{v_2 - v_1}{t_1 + t_2} \right| \]