A self-propelled vehicle of mass m, whose engine delivers a constant power P, has an acceleration `a = (P//mv)`. (Assume that there is no friction). In order to increase its velocity from `v_(1)` to `v_(2)`, the distan~e it has to travel will be:

Options

A. `(3P)/(m)(v_(2)^(2)-v_(1)^(2))`
B. `(m)/(3P)(v_(2)-v_(1))`
C. `(m)/(3P)(v_(2)^(3)-v_(1)^(3))`
D. `(m)/(3P)(v_(2)^(2)-v_(1)^(2))`

Answer

To solve the problem of finding the distance a self-propelled vehicle must travel to increase its velocity from \( v_1 \) to \( v_2 \), given that its engine delivers a constant power \( P \) and has an acceleration defined by \( a = \frac{P}{mv} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The acceleration \( a \) can be expressed in terms of velocity \( v \) and time \( t \) as:
\[ a = \frac{dv}{dt} \]
Given that \( a = \frac{P}{mv} \), we can equate the two expressions for acceleration.

2. **Rearranging the Equation**:
We can rearrange the equation:
\[ \frac{dv}{dt} = \frac{P}{mv} \]
Now, we can multiply both sides by \( dt \) and \( v \):
\[ v \, dv = \frac{P}{m} \, dt \]

3. **Using the Chain Rule**:
We can express \( dt \) in terms of \( dx \) (distance):
\[ dt = \frac{dx}{v} \]
Substituting this into our equation gives:
\[ v \, dv = \frac{P}{m} \frac{dx}{v} \]
Rearranging this leads to:
\[ v^2 \, dv = \frac{P}{m} \, dx \]

4. **Integrating Both Sides**:
Now we can integrate both sides. The left side will be integrated with respect to \( v \) from \( v_1 \) to \( v_2 \), and the right side will be integrated with respect to \( x \) from \( x_1 \) to \( x_2 \):
\[ \int_{v_1}^{v_2} v^2 \, dv = \frac{P}{m} \int_{x_1}^{x_2} dx \]

5. **Calculating the Integrals**:
The integral of \( v^2 \) is:
\[ \frac{v^3}{3} \bigg|_{v_1}^{v_2} = \frac{v_2^3}{3} - \frac{v_1^3}{3} \]
The integral of \( dx \) is simply:
\[ x \bigg|_{x_1}^{x_2} = x_2 - x_1 \]

6. **Setting Up the Equation**:
Now we can set up the equation:
\[ \frac{v_2^3 - v_1^3}{3} = \frac{P}{m} (x_2 - x_1) \]

7. **Solving for Distance**:
Rearranging gives us the distance traveled:
\[ x_2 - x_1 = \frac{m}{3P} (v_2^3 - v_1^3) \]

### Final Result:
Thus, the distance \( d \) that the vehicle must travel to increase its velocity from \( v_1 \) to \( v_2 \) is:
\[ d = \frac{m}{3P} (v_2^3 - v_1^3) \]