Step 1: Understanding the Concept:
Acceleration is defined as the rate of change of velocity with respect to time.
On a velocity-time (\(v-t\)) graph, the slope of the curve at any point gives the instantaneous acceleration.
If the relationship between velocity and time is linear (\( v = mt + c \)), the slope is constant, meaning the acceleration is uniform.
Step 2: Key Formula or Approach:
The formula for instantaneous acceleration is the derivative of velocity:
\[ a = \frac{dv}{dt} \]
For a linear equation \( v = kt \), the acceleration is simply the coefficient \( k \).
Step 3: Detailed Explanation:
1. We are given the equation for velocity: \( v = 4t \).
2. Differentiate the velocity with respect to time to find acceleration:
\[ a = \frac{d}{dt}(4t) \]
3. Since the derivative of \( t \) with respect to \( t \) is 1:
\[ a = 4 \times \frac{dt}{dt} = 4 \text{ m/s}^2 \]
4. Observations:
- The resulting value is a constant (it does not depend on \( t \)).
- This implies the particle is moving with uniform acceleration.
- If the equation had been \( v = 4t^2 \), the acceleration would have been \( 8t \), which is variable.
Step 4: Final Answer:
The acceleration is constant and equals 4 m/s\(^2\).
This corresponds to Option (B).