The speed v reached by a car of mass m in travelling a distance x, driven with constant power P, is given by

Options

A. (a) `v=(3xP)/(m)`
B. (b) `v=((3xP)/(m))^(1//2)`
C. (c) `v=((3xP)/(m))^(1//3)`
D. (d) `v=((3xP)/(m))^2`

Explanation

To solve the problem of finding the speed \( v \) reached by a car of mass \( m \) traveling a distance \( x \) while being driven with constant power \( P \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between power, force, and velocity**:
The power \( P \) delivered by the car can be expressed as:
\[ P = F \cdot v \]
where \( F \) is the force acting on the car and \( v \) is its velocity.

2. **Express force in terms of mass and acceleration**:
The force \( F \) can also be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( a \) is the acceleration of the car.

3. **Relate acceleration to velocity and distance**:
We know that acceleration \( a \) can be expressed as:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \cdot v \]
Thus, we can rewrite the power equation as:
\[ P = m \cdot a \cdot v = m \cdot \left(\frac{dv}{dx} \cdot v\right) \cdot v = m \cdot v \cdot \frac{dv}{dx} \]

4. **Rearranging the equation**:
Rearranging the equation gives:
\[ P = m \cdot v \cdot \frac{dv}{dx} \]
From this, we can isolate \( \frac{dv}{dx} \):
\[ \frac{dv}{dx} = \frac{P}{m \cdot v} \]

5. **Integrate both sides**:
We need to integrate both sides. The left side will be integrated with respect to \( v \) from \( 0 \) to \( v \) and the right side with respect to \( x \) from \( 0 \) to \( x \):
\[ \int_0^v v \, dv = \int_0^x \frac{P}{m} \, dx \]

6. **Perform the integration**:
The left side integrates to:
\[ \frac{v^2}{2} \bigg|_0^v = \frac{v^2}{2} \]
The right side integrates to:
\[ \frac{P}{m} \cdot x \bigg|_0^x = \frac{P}{m} \cdot x \]

7. **Set the two integrals equal**:
Thus, we have:
\[ \frac{v^2}{2} = \frac{P}{m} \cdot x \]

8. **Solve for \( v \)**:
Multiplying both sides by 2 gives:
\[ v^2 = \frac{2Px}{m} \]
Taking the square root of both sides results in:
\[ v = \sqrt{\frac{2Px}{m}} \]

9. **Final expression**:
To express \( v \) in terms of \( P \), \( x \), and \( m \):
\[ v = \sqrt{\frac{2Px}{m}} \]

### Final Answer:
The speed \( v \) reached by a car of mass \( m \) traveling a distance \( x \) driven with constant power \( P \) is given by:
\[ v = \sqrt{\frac{2Px}{m}} \]