A particle starts from rest with a time varying acceleration `a=(2t-4)`. Here `t` is in second and `a` in `m//s^(2)`
The velocity time graph of the particle is

Options

A. parabola passing through origin
B. straight line not passing through origin
C. parabola not passing through origin
D. straight line passing through origin

Answer

To find the velocity-time graph of a particle with a time-varying acceleration given by \( a = 2t - 4 \), we can follow these steps:

### Step 1: Understand the relationship between acceleration and velocity
Acceleration \( a \) is defined as the rate of change of velocity with respect to time:
\[ a = \frac{dv}{dt} \]
Given that \( a = 2t - 4 \), we can write:
\[ \frac{dv}{dt} = 2t - 4 \]

### Step 2: Rearrange the equation
We can rearrange the equation to separate variables:
\[ dv = (2t - 4) dt \]

### Step 3: Integrate both sides
Next, we integrate both sides. The left side integrates to velocity \( v \), and the right side integrates with respect to time \( t \):
\[ \int dv = \int (2t - 4) dt \]
This gives us:
\[ v = \int (2t - 4) dt \]

### Step 4: Perform the integration
The integral of \( 2t \) is \( t^2 \) and the integral of \( -4 \) is \( -4t \):
\[ v = t^2 - 4t + C \]
where \( C \) is the constant of integration.

### Step 5: Determine the constant of integration
Since the particle starts from rest, the initial velocity \( v(0) = 0 \):
\[ v(0) = 0^2 - 4(0) + C = 0 \implies C = 0 \]
Thus, the equation simplifies to:
\[ v = t^2 - 4t \]

### Step 6: Analyze the velocity equation
The equation \( v = t^2 - 4t \) is a quadratic equation in \( t \). To find the nature of the graph, we can rewrite it as:
\[ v = t(t - 4) \]
This indicates that the velocity is zero at \( t = 0 \) and \( t = 4 \).

### Step 7: Determine the shape of the graph
Since this is a quadratic equation that opens upwards (the coefficient of \( t^2 \) is positive), the graph is a parabola. It passes through the origin (0,0) and also intersects the time axis at \( t = 4 \).

### Conclusion
The velocity-time graph of the particle is a parabola that opens upwards, passing through the origin and intersecting the time axis at \( t = 4 \).

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