A particle start from rest, with uniform acceleration at time `t`, choose the correct option

Options

A.
B.
C.
D.

Answer

To solve the problem step by step, we will analyze the motion of a particle starting from rest with uniform acceleration.

### Step 1: Understand the Given Information
- The particle starts from rest, which means the initial velocity \( u = 0 \).
- The particle has uniform (constant) acceleration, which we will denote as \( a \).

### Step 2: Write the Equation for Velocity
The relationship between acceleration, velocity, and time can be expressed as:
\[ a = \frac{dv}{dt} \]
Since acceleration is constant, we can integrate this equation. Rearranging gives:
\[ dv = a \, dt \]
Integrating both sides:
\[ \int dv = \int a \, dt \]
This results in:
\[ v = at + C \]
Given that the initial velocity \( u = 0 \) when \( t = 0 \), we find that \( C = 0 \). Therefore, the equation for velocity becomes:
\[ v = at \]

### Step 3: Write the Equation for Position
The velocity can also be expressed as the derivative of position with respect to time:
\[ v = \frac{dx}{dt} \]
Substituting our expression for velocity:
\[ \frac{dx}{dt} = at \]
Now, we integrate this equation:
\[ dx = at \, dt \]
Integrating both sides gives:
\[ \int dx = \int at \, dt \]
This results in:
\[ x = \frac{at^2}{2} + C \]
Again, since the particle starts from the origin (initial position \( x = 0 \) when \( t = 0 \)), we find that \( C = 0 \). Thus, the equation for position becomes:
\[ x = \frac{at^2}{2} \]

### Step 4: Analyze the Graphs
1. **Acceleration-Time Graph**: Since acceleration is constant, the graph will be a horizontal line at \( a \).
2. **Velocity-Time Graph**: The graph will be a straight line with a slope equal to \( a \) and passing through the origin (0,0).
3. **Position-Time Graph**: The graph will be a parabola opening upwards, represented by the equation \( x = \frac{at^2}{2} \).

### Conclusion
Based on the analysis:
- Acceleration is constant.
- Velocity is a linear function of time, passing through the origin.
- Position is a quadratic function of time, represented by a parabolic graph.

### Final Answer
All options regarding acceleration, velocity, and position are correct.