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For a particle moving in straight line w...

For a particle moving in straight line with increasing speed the appropriate sign of acceleration a and velocity v can be:

A

`agt0` and `vgt0`

B

`alt0` and `vlt0`

C

`agt0` and `vlt0`

D

`alt0` and `vgt0`

Text Solution

AI Generated Solution

The correct Answer is:
To determine the appropriate signs of acceleration \( a \) and velocity \( v \) for a particle moving in a straight line with increasing speed, we can analyze the relationship between velocity and acceleration. ### Step-by-Step Solution: 1. **Understanding Velocity and Acceleration**: - Velocity (\( v \)) indicates the speed and direction of the particle. - Acceleration (\( a \)) indicates the rate of change of velocity. It can be positive (speeding up) or negative (slowing down). 2. **Case 1: Both Velocity and Acceleration are Positive**: - If \( v > 0 \) and \( a > 0 \), the particle is moving in the positive direction and speeding up. - This scenario leads to an increase in speed. - **Conclusion**: This case is valid. 3. **Case 2: Both Velocity and Acceleration are Negative**: - If \( v < 0 \) and \( a < 0 \), the particle is moving in the negative direction and speeding up (in the negative direction). - This also results in an increase in speed (though the speed is considered negative). - **Conclusion**: This case is also valid. 4. **Case 3: Positive Acceleration and Negative Velocity**: - If \( v < 0 \) and \( a > 0 \), the particle is moving in the negative direction but experiencing positive acceleration. - This means the particle is slowing down (decelerating) in the negative direction, which results in a decrease in speed. - **Conclusion**: This case is invalid. 5. **Case 4: Negative Acceleration and Positive Velocity**: - If \( v > 0 \) and \( a < 0 \), the particle is moving in the positive direction but experiencing negative acceleration. - This means the particle is slowing down, which results in a decrease in speed. - **Conclusion**: This case is also invalid. ### Final Conclusion: The valid cases where the particle is moving in a straight line with increasing speed are: - Both \( a \) and \( v \) are positive. - Both \( a \) and \( v \) are negative. Thus, the correct options are: - Option A: \( a > 0 \), \( v > 0 \) - Option B: \( a < 0 \), \( v < 0 \)

To determine the appropriate signs of acceleration \( a \) and velocity \( v \) for a particle moving in a straight line with increasing speed, we can analyze the relationship between velocity and acceleration. ### Step-by-Step Solution: 1. **Understanding Velocity and Acceleration**: - Velocity (\( v \)) indicates the speed and direction of the particle. - Acceleration (\( a \)) indicates the rate of change of velocity. It can be positive (speeding up) or negative (slowing down). ...
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