A particle is moving on the x-axis. If initial velocity `ugt0`, and acceleration `alt0` and a is constant then number of possible calues of the time for which the particle is at distance `d` from its starting point may be:

Options

A. one
B. two
C. three
D. four

Explanation

To solve the problem, we need to analyze the motion of the particle on the x-axis given its initial conditions. The particle has an initial velocity \( u > 0 \) and a constant negative acceleration \( a < 0 \).

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle starts with an initial velocity \( u \) and is decelerating due to the negative acceleration \( a \). This means the particle will eventually stop and then start moving in the opposite direction.

2. **Equation of Motion**:
- The equation of motion for the particle can be expressed as:
\[ x(t) = ut + \frac{1}{2} a t^2 \]
- Here, \( x(t) \) is the position of the particle at time \( t \).

3. **Finding the Distance**:
- We need to find the times \( t \) when the particle is at a distance \( d \) from its starting point (i.e., \( x(t) = d \)).
- Setting the equation equal to \( d \):
\[ d = ut + \frac{1}{2} a t^2 \]
- Rearranging gives us a quadratic equation:
\[ \frac{1}{2} a t^2 + ut - d = 0 \]

4. **Analyzing the Quadratic Equation**:
- This is a standard quadratic equation of the form \( At^2 + Bt + C = 0 \), where:
- \( A = \frac{1}{2} a \)
- \( B = u \)
- \( C = -d \)
- The number of solutions (or times \( t \)) depends on the discriminant \( D \) of the quadratic equation:
\[ D = B^2 - 4AC = u^2 - 2ad \]

5. **Determining the Number of Solutions**:
- The number of possible values of \( t \) can be determined by the value of the discriminant \( D \):
- If \( D > 0 \): There are **two distinct solutions** (the particle reaches \( d \) and then returns).
- If \( D = 0 \): There is **one solution** (the particle stops at \( d \)).
- If \( D < 0 \): There are **no real solutions** (the particle never reaches \( d \)).
- Since \( a < 0 \) (deceleration), we need to ensure that \( u^2 - 2ad \) can be positive, zero, or negative depending on the values of \( u \), \( a \), and \( d \).

6. **Conclusion**:
- The possible values of time for which the particle is at distance \( d \) from its starting point can be:
- **1 time** (if it stops at \( d \)),
- **2 times** (if it reaches \( d \) and returns),
- **3 times** (if it goes past \( d \) and returns back).
- Thus, the total number of possible values of time is **1, 2, or 3**.