Which graph represents the uniform acceleration

Options

A.
B.
C.
D.

Answer

To determine which graph represents uniform acceleration, we can follow these steps:

### Step 1: Understand Uniform Acceleration
Uniform acceleration means that the acceleration is constant over time. This implies that the velocity of the object changes at a constant rate.

### Step 2: Relate Acceleration to Velocity
Since acceleration (a) is defined as the rate of change of velocity (v) with respect to time (t), we can express this mathematically as:
\[ a = \frac{dv}{dt} \]
For uniform acceleration, this means that \( a \) is a constant value (let's denote it as \( k \)):
\[ \frac{dv}{dt} = k \]

### Step 3: Integrate to Find Velocity
Integrating both sides with respect to time gives us the velocity as a function of time:
\[ dv = k \, dt \]
Integrating:
\[ v = kt + C \]
where \( C \) is the constant of integration, which represents the initial velocity.

### Step 4: Relate Velocity to Displacement
Now, we can relate velocity to displacement (s) using the formula:
\[ v = \frac{ds}{dt} \]
Substituting the expression for \( v \):
\[ \frac{ds}{dt} = kt + C \]

### Step 5: Integrate to Find Displacement
Integrating this equation with respect to time gives us the displacement:
\[ ds = (kt + C) \, dt \]
Integrating:
\[ s = \frac{k}{2}t^2 + Ct + D \]
where \( D \) is another constant of integration.

### Step 6: Identify the Form of the Displacement Equation
The displacement equation can be simplified to:
\[ s = \frac{k}{2}t^2 + Ct + D \]
This is a quadratic equation in \( t \), which means that the graph of displacement (s) versus time (t) will be a parabola.

### Step 7: Determine the Shape of the Graph
Since the coefficient of \( t^2 \) is positive (\( \frac{k}{2} \)), the parabola opens upwards. This indicates that the object is experiencing uniform acceleration.

### Step 8: Compare with Given Options
Now, we compare the derived graph (upward-opening parabola) with the options provided in the question. The correct option will be the one that shows an upward-opening parabola.

### Conclusion
The graph that represents uniform acceleration is the one that is a parabola opening upwards.

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