A car drives along a straight level frictionless road by an engine delivering constant power. Then velocity is directly proportional to

Options

A. (a) `sqrtt`
B. (b) `1/sqrtt`
C. (c) `sqrtt`
D. (d) None of the above

Explanation

To solve the problem, we need to analyze the relationship between the velocity of the car and the time when the engine is delivering constant power.

### Step-by-Step Solution:

1. **Understanding Power**:
The power \( P \) delivered by the engine is defined as the product of force \( F \) and velocity \( v \):
\[ P = F \cdot v \]

2. **Relating Force to Mass and Acceleration**:
According to Newton's second law, the force can also be expressed as:
\[ F = m \cdot a \]
where \( m \) is the mass of the car and \( a \) is its acceleration.

3. **Acceleration as a Function of Velocity**:
Acceleration \( a \) can be defined as the rate of change of velocity:
\[ a = \frac{dv}{dt} \]
Substituting this into the force equation gives:
\[ F = m \cdot \frac{dv}{dt} \]

4. **Substituting Force in Power Equation**:
Now, substituting \( F \) into the power equation:
\[ P = m \cdot \frac{dv}{dt} \cdot v \]

5. **Rearranging the Equation**:
Rearranging the equation gives:
\[ P = m \cdot v \cdot \frac{dv}{dt} \]
This can be rewritten as:
\[ v \, dv = \frac{P}{m} \, dt \]

6. **Integrating Both Sides**:
Now we integrate both sides. The left side integrates with respect to \( v \) and the right side with respect to \( t \):
\[ \int v \, dv = \int \frac{P}{m} \, dt \]
This results in:
\[ \frac{v^2}{2} = \frac{P}{m} t + C \]
where \( C \) is the constant of integration.

7. **Solving for Velocity**:
Rearranging the equation to solve for \( v \):
\[ v^2 = \frac{2P}{m} t + 2C \]
Taking the square root gives:
\[ v = \sqrt{\frac{2P}{m} t + 2C} \]

8. **Identifying Proportionality**:
If we consider the term \( \sqrt{t} \) in the equation, we can see that as \( t \) increases, \( v \) is directly proportional to \( \sqrt{t} \).

Thus, we conclude that:
\[ v \propto t^{1/2} \]

### Final Answer:
The velocity \( v \) is directly proportional to the square root of time \( t \) when the engine delivers constant power.