A police party is chasing a dacoit in a jeep which is moving at a constant speed `v`. The dacoit is on a motor cycle. When he is at a distance `x` from the jeep, he accelerates from rest at a constant rate `alpha`. Which of the following relations is true, if the police is able to catch the dacoit ?

Options

A. `v^(2) lt ax`
B. `v^(2) lt 2ax`
C. `v^(2) ge 2ax`
D. `v^(2) = ax`

Explanation

To solve the problem step by step, we need to analyze the motion of both the police jeep and the dacoit's motorcycle.

### Step 1: Understanding the Motion of the Dacoit
The dacoit starts from rest and accelerates at a constant rate \(\alpha\). The distance he travels after time \(t\) can be described by the equation of motion:
\[ s_d = \frac{1}{2} \alpha t^2 \]
where \(s_d\) is the distance traveled by the dacoit.

### Step 2: Understanding the Motion of the Police Jeep
The police jeep is moving at a constant speed \(v\). The distance traveled by the police jeep in the same time \(t\) is given by:
\[ s_p = vt \]
where \(s_p\) is the distance traveled by the police.

### Step 3: Setting Up the Equation
At the moment the police catch the dacoit, the distance traveled by the dacoit plus the initial distance \(x\) should equal the distance traveled by the police jeep. Therefore, we can set up the equation:
\[ s_d + x = s_p \]
Substituting the expressions for \(s_d\) and \(s_p\), we have:
\[ \frac{1}{2} \alpha t^2 + x = vt \]

### Step 4: Rearranging the Equation
Rearranging the equation gives us:
\[ \frac{1}{2} \alpha t^2 - vt + x = 0 \]
This is a quadratic equation in \(t\).

### Step 5: Applying the Condition for Real Roots
For the police to catch the dacoit, this quadratic equation must have real solutions for \(t\). This means that the discriminant \(D\) of the quadratic equation must be greater than or equal to zero:
\[ D = b^2 - 4ac \]
In our case, \(a = \frac{1}{2} \alpha\), \(b = -v\), and \(c = x\). Thus, the discriminant becomes:
\[ D = (-v)^2 - 4 \left(\frac{1}{2} \alpha\right) x = v^2 - 2\alpha x \]
Setting the discriminant greater than or equal to zero gives:
\[ v^2 - 2\alpha x \geq 0 \]

### Step 6: Final Relation
From the above inequality, we can conclude:
\[ v^2 \geq 2\alpha x \]

### Conclusion
The relation that must hold true for the police to catch the dacoit is:
\[ v^2 \geq 2\alpha x \]