A particle moves along a straight line path. After some time it comes to rest. The motion is with constant acceleration whose direction with respect to the direction of velocity is :

Options

A. positive throughout motion
B. negative throughout motion
C. first positive then negative
D. first negative then negative

Answer

To solve the problem, we need to analyze the motion of a particle that moves along a straight line and comes to rest under constant acceleration. Here's a step-by-step solution:

### Step 1: Understand the Motion
The particle is initially moving with some velocity (let's denote it as \( V_i \)) and eventually comes to rest, meaning its final velocity (\( V_f \)) is 0.

### Step 2: Use the Equation of Motion
Since the particle is moving with constant acceleration, we can use the equation of motion:
\[ V_f = V_i + a \cdot t \]
where:
- \( V_f \) = final velocity (0, since the particle comes to rest)
- \( V_i \) = initial velocity
- \( a \) = acceleration
- \( t \) = time

### Step 3: Rearranging the Equation
Rearranging the equation gives us:
\[ 0 = V_i + a \cdot t \]
This can be rewritten as:
\[ a \cdot t = -V_i \]
From this, we can express acceleration as:
\[ a = -\frac{V_i}{t} \]

### Step 4: Analyze the Sign of Acceleration
Since \( V_i \) is positive (the particle is moving in the positive direction), and \( t \) is also positive (time cannot be negative), the acceleration \( a \) is negative. This indicates that the acceleration is in the opposite direction to the initial velocity.

### Step 5: Conclusion
The direction of acceleration is opposite to the direction of velocity. Therefore, the acceleration is negative with respect to the direction of the velocity.

### Final Answer
The direction of acceleration with respect to the direction of velocity is negative.

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