A particle is moving in `x-y` plane with `y=x/2` and `V_(x)=4-2t`. The displacement versus time graph of the particle would be

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Answer

To solve the problem step by step, we will analyze the motion of the particle in the x-y plane and derive the displacement versus time graph.

### Step 1: Understand the given equations
The particle is moving in the x-y plane with the equation for y given as:
\[ y = \frac{x}{2} \]
And the velocity in the x-direction is given as:
\[ V_x = 4 - 2t \]

### Step 2: Relate velocity to displacement
The velocity \( V_x \) can be expressed as the derivative of displacement \( x \) with respect to time \( t \):
\[ V_x = \frac{dx}{dt} \]
Thus, we can write:
\[ \frac{dx}{dt} = 4 - 2t \]

### Step 3: Rearrange the equation for integration
Rearranging the equation gives us:
\[ dx = (4 - 2t) dt \]

### Step 4: Integrate both sides
Now, we will integrate both sides. The left side integrates to \( x \), and the right side can be integrated term by term:
\[ \int dx = \int (4 - 2t) dt \]
This results in:
\[ x = 4t - t^2 + C \]
where \( C \) is the constant of integration.

### Step 5: Determine the constant of integration
Assuming the particle starts at the origin when \( t = 0 \), we have:
\[ x(0) = 0 \implies 0 = 4(0) - (0)^2 + C \implies C = 0 \]
Thus, the equation simplifies to:
\[ x = 4t - t^2 \]

### Step 6: Analyze the equation of motion
The equation \( x = 4t - t^2 \) is a quadratic equation in terms of \( t \). This represents a parabola opening downwards.

### Step 7: Determine the displacement versus time graph
Since \( x \) is a quadratic function of \( t \), the displacement versus time graph will be a parabola. The vertex of this parabola can be found using the formula \( t = -\frac{b}{2a} \) where \( a = -1 \) and \( b = 4 \):
\[ t = -\frac{4}{2(-1)} = 2 \]
At \( t = 2 \):
\[ x = 4(2) - (2)^2 = 8 - 4 = 4 \]
Thus, the vertex of the parabola is at the point (2, 4).

### Conclusion
The displacement versus time graph of the particle is a downward-opening parabola with its vertex at (2, 4).