The velocity of a body starting from rest is directly proportional to time. Which of these is uniform?

Options

A. Velocity
B. Acceleration
C. Both velocity and acceleration
D. Neither velocity nor acceleration

Answer

To solve the question, "The velocity of a body starting from rest is directly proportional to time. Which of these is uniform?" we can follow these steps:

### Step 1: Understand the relationship between velocity and time
The problem states that the velocity (V) of a body starting from rest is directly proportional to time (t). This can be expressed mathematically as:
\[ V \propto t \]
This means that as time increases, the velocity increases linearly.

### Step 2: Express the relationship mathematically
Since velocity is directly proportional to time, we can write:
\[ V = k \cdot t \]
where \( k \) is a constant of proportionality.

### Step 3: Determine the acceleration
Acceleration (a) is defined as the rate of change of velocity with respect to time. Mathematically, this is given by:
\[ a = \frac{dV}{dt} \]
Substituting the expression for velocity, we have:
\[ a = \frac{d(k \cdot t)}{dt} \]

### Step 4: Calculate the derivative
Calculating the derivative:
\[ a = k \cdot \frac{d(t)}{dt} = k \]
Since \( k \) is a constant, the acceleration \( a \) is also a constant.

### Step 5: Conclusion
Since the acceleration is constant, we can conclude that the body is undergoing uniform acceleration. Therefore, the answer to the question is that the acceleration is uniform.

### Final Answer
The uniform quantity in this scenario is **acceleration**.

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