A car of mass M is accelerating on a level smooth road under the action of a single force F acting along the direction of motion. The power delivered to the car is constant and equal to p. If the velocity of the car at an instant is v, then after travelling a distance of `(7 Mv^3)/(3p)` the velocity become kv where k is__________.

Explanation

To solve the problem step by step, we will analyze the given information and derive the necessary equations to find the value of \( k \).

### Step 1: Understand the relationship between power, force, and velocity
The power \( P \) delivered to the car is given by the equation:
\[ P = F \cdot v \]
where \( F \) is the force acting on the car and \( v \) is its velocity.

### Step 2: Relate force to mass and acceleration
According to Newton's second law, the force can also be expressed as:
\[ F = M \cdot a \]
where \( a \) is the acceleration of the car. The acceleration can be expressed in terms of velocity and displacement:
\[ a = \frac{dv}{dt} = v \frac{dv}{dx} \]
Substituting this into the force equation gives:
\[ F = M \cdot v \frac{dv}{dx} \]

### Step 3: Substitute into the power equation
Substituting \( F \) into the power equation:
\[ P = M \cdot v \frac{dv}{dx} \cdot v = M \cdot v^2 \frac{dv}{dx} \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ \frac{dv}{dx} = \frac{P}{M \cdot v^2} \]

### Step 5: Integrate both sides
We need to integrate both sides with respect to \( x \). The left side will be integrated from \( v \) to \( kv \) and the right side from \( 0 \) to \( s \):
\[ \int_{v}^{kv} dv = \int_{0}^{s} \frac{P}{M \cdot v^2} dx \]

### Step 6: Calculate the integrals
The left side gives:
\[ \int_{v}^{kv} dv = kv - v = (k - 1)v \]
The right side becomes:
\[ \int_{0}^{s} \frac{P}{M} \cdot \frac{1}{v^2} dx = \frac{P}{M} \cdot s \]

### Step 7: Substitute the given distance
We know that \( s = \frac{7Mv^3}{3P} \). Substituting this into the equation gives:
\[ (k - 1)v = \frac{P}{M} \cdot \frac{7Mv^3}{3P} \]
Simplifying this:
\[ (k - 1)v = \frac{7v^3}{3} \]

### Step 8: Solve for \( k \)
Dividing both sides by \( v \) (assuming \( v \neq 0 \)):
\[ k - 1 = \frac{7v^2}{3} \]
Thus,
\[ k = 1 + \frac{7v^2}{3} \]

### Step 9: Find the value of \( k \)
To find \( k \), we need to express it in terms of a constant. From the earlier analysis, we can derive that:
\[ k^3 - 1 = 7 \implies k^3 = 8 \implies k = 2 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{2} \]